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The Two-Parametric Mittag-Leffler Function

  • Rudolf Gorenflo
  • Anatoly A. Kilbas
  • Francesco Mainardi
  • Sergei V. Rogosin
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function E α, β (z) (see ( 1.0.3)), which is the most straightforward generalization of the classical Mittag-Leffler function E α (z) (see ( 3.1.1)).

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function E α, β (z) (see ( 1.0.3)), which is the most straightforward generalization of the classical Mittag-Leffler function E α (z) (see ( 3.1.1)). As in the previous chapter, the material can be formally divided into two parts. At first, starting from the basic definition of the Mittag-Leffler function as a power series, we discover that, for the first parameter α with positive real part and any complex value of the second parameter β, the function E α, β (z) is an entire function of the complex variable z. Therefore we discuss in the first part the (analytic) properties of the two-parametric Mittag-Leffler function as an entire function. Namely, we calculate its order and type, present a number of formulas relating the two-parametric Mittag-Leffler function to elementary and special functions as well as recurrence relations and differentiation formulas, introduce some useful integral representations and discuss asymptotics and the distribution of zeros of the considered function. An extension of the two-parametric Mittag-Leffler function to values of the first parameter α with non-positive real part is given here too.

It is well-known that in current applications the properties of the two-parametric Mittag-Leffler function of a real variable are often used. Thus, we collect in the second part (Sect. 4.10) results of this type. They concern integral representations and integral transforms of the two-parametric Mittag-Leffler function of a real variable, the complete monotonicity property, and relations to the fractional calculus. People working in applications can, at first reading, omit some of the deeper mathematical material (that from Sects. 4.44.9, say).

4.1 Series Representation and Properties of Coefficients

The Mittag-Leffler type function (or the two-parametric Mittag-Leffler function)
$$\displaystyle{ E_{\alpha,\beta }(z) =\sum _{ k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha k+\beta )}\quad (\alpha > 0,\;\beta \in \mathbb{C}) }$$
(4.1.1)
generalizes the classical Mittag-Leffler function
$$\displaystyle{E_{\alpha }(z) =\sum _{ k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha k + 1)}\quad (\alpha > 0).}$$
We have \(E_{\alpha,1}(z) = E_{\alpha }(z)\). For any \(\alpha,\beta \in \mathbb{C}\), Re α > 0, the function (4.1.1) is an entire function of order ρ = 1∕(Re α) and type 1.
Indeed, let us consider the slightly more general function
$$\displaystyle{ E_{\alpha,\beta }(\sigma ^{\alpha }\,z) =\sum _{ k=0}^{\infty }\frac{\left (\sigma ^{\alpha }\,z\right )^{k}} {\varGamma (\alpha k+\beta )}, }$$
(4.1.2)
i.e. take the coefficients in the form
$$\displaystyle{ c_{k} = \frac{\sigma ^{\alpha k}} {\varGamma \left (\alpha k+\beta \right )}\quad (k = 0,\,1,\,2,\ldots ), }$$
(4.1.3)
where 0 < Re α < +, 0 < σ < + is an arbitrary real constant and β is a complex parameter. By using Stirling’s formula (see, e.g., [Bat-1, 1.18 (3)]) we have
$$\displaystyle{ \varGamma \left (\alpha k+\beta \right ) = \sqrt{2\pi }\left (\alpha k\right )^{\alpha k+\beta -\frac{1} {2} }\,\mathrm{e}^{-\alpha k}\,[1 + o(1)],\quad k \rightarrow \infty }$$
(4.1.4)
and, consequently, for the sequence \(\{c_{k}\}_{0}^{\infty }\) we immediately obtain
$$\displaystyle{ \lim _{k\rightarrow \infty } \frac{k\log k} {\log \frac{1} {\vert c_{k}\vert }} = \frac{1} {(\mathrm{Re}\,\alpha )} =\rho,\quad \lim _{k\rightarrow \infty }k\vert c_{k}\vert ^{\rho /k} =\mathrm{ e}\rho \sigma. }$$
(4.1.5)
According to a well known theorem in the theory of entire functions (see, e.g., formulas (B.2.3) and (B.2.4)), the function (4.1.2) has order ρ = 1∕α and type σ for any β. Therefore, the two-parametric Mittag-Leffler function (4.1.1) has order ρ = 1∕(Re α) and type 1 for any value of the parameter \(\beta \in \mathbb{C}\).

4.2 Explicit Formulas: Relations to Elementary and Special Functions

Using definition (4.1.1) we obtain a number of formulas relating the two-parametric Mittag-Leffler function E α, β to elementary functions (see, e.g. [HaMaSa11])
$$\displaystyle\begin{array}{rcl} E_{1,1}(z) =\mathrm{ e}^{z},\quad E_{ 1,2}(z) = \frac{\mathrm{e}^{z} - 1} {z},& &{}\end{array}$$
(4.2.1)
$$\displaystyle\begin{array}{rcl} E_{2,1}(z) =\cosh \sqrt{z},\quad E_{2,2}(z) = \frac{\sinh \sqrt{z}} {\sqrt{z}}.& &{}\end{array}$$
(4.2.2)
Some extra formulas, for other special values of parameters, are presented as exercises at the end of the chapter.
Here we also mention two recurrence relations for the function (4.1.1).
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) = \frac{1} {\varGamma (\beta )} + zE_{\alpha,\beta +\alpha }\left (z\right ),& &{}\end{array}$$
(4.2.3)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\beta E_{\alpha,\beta +1}(z) +\alpha z\, \frac{\mathrm{d}} {\mathrm{d}z}E_{\alpha,\beta +1}(z).& &{}\end{array}$$
(4.2.4)

Now we present some other relations of the two-parametric Mittag-Leffler function to certain special functions introduced by different authors.

The hyperbolic functions of order n, denoted by h r (z, n), are defined, e.g., in [Bat-3, 18.2 (2)]. Their series representations relate these functions to the two-parametric Mittag-Leffler function (see [Bat-3, 18.2 (16)]):
$$\displaystyle{ h_{r}(z,n) =\sum \limits _{ k=0}^{\infty } \frac{z^{\mathit{nk}+r-1}} {(\mathit{nk} + r - 1)!} = z^{r-1}E_{ n,r}(z^{n}),\;\;\;r = 1,2,\ldots }$$
(4.2.5)
The trigonometric functions of order n, denoted by k r (z, n), are defined, e.g., in [Bat-3, 18.2 (18)]. With \(\lambda = \mbox{ exp}\left \{ \frac{\pi i} {n}\right \}\) the functions k r (z, n) and h r (z, n) are related by:
$$\displaystyle{k_{r}(z,n) =\lambda ^{1-r}h_{ r}(\lambda z,n),}$$
from which it follows
$$\displaystyle{ k_{r}(z,n) =\sum \limits _{ j=0}^{\infty }\frac{(-1)^{j}z^{\mathit{nj}+r-1}} {(\mathit{nj} + r - 1)!} = z^{r-1}E_{ n,r}(-z^{n}),\;\;\;r = 1,2,\ldots }$$
(4.2.6)
The relation to the complementary error function is also well-known (see [MatHau08, pp. 80–81], [Dzh66, p. 297]):
$$\displaystyle{ E_{\frac{1} {2},1}(z) =\sum \limits _{ k=0}^{\infty } \frac{z^{k}} {\varGamma \left (\frac{k} {2} + 1\right )} =\mathrm{ e}^{z^{2} }\mathrm{erfc}(-z), }$$
(4.2.7)
where erfc is complementary to the error function erf:
$$\displaystyle{\mathrm{erfc}(z):= \frac{2} {\sqrt{\pi }}\int _{z}^{\infty }\mathrm{e}^{-u^{2} }\mathrm{d}u = 1 -\mathrm{erf}(z),\;z \in \mathbb{C}.}$$
The Miller–Ross function is defined as follows (see [MilRos93]):
$$\displaystyle{ E_{t}(\nu,a) = t^{\nu }\sum \limits _{k=0}^{\infty } \frac{(\mathit{at})^{k}} {\varGamma \left (\nu +k + 1\right )} = t^{\nu }E_{1,\nu +1}(\mathit{at}). }$$
(4.2.8)
The Rabotnov function is represented as follows (see [RaPaZv69]):
$$\displaystyle{ R_{\alpha }(\beta,a) = t^{\alpha }\sum \limits _{k=0}^{\infty } \frac{\beta ^{k}t^{k(\alpha +1)}} {\varGamma \left ((1+\alpha )(k + 1)\right )} = t^{\alpha }E_{\alpha +1,\alpha +1}(\beta t^{\alpha +1}). }$$
(4.2.9)

4.3 Differential and Recurrence Relations

The following differentiation formula is an immediate consequence of the definition of the two-parametric Mittag-Leffler function (4.1.1)
$$\displaystyle{ \left ( \frac{\mathrm{d}} {\mathrm{d}z}\right )^{m}[z^{\beta -1}E_{\alpha,\beta }(z^{\alpha })] = z^{\beta -m-1}E_{\alpha,\beta -m}(z^{\alpha })\quad (m \geq 1). }$$
(4.3.1)
We consider now some corollaries of formula (4.3.1). Let \(\alpha = m/n,\ (m,\ n = 1,\ 2,\ \ldots )\) in (4.3.1). Then
$$\displaystyle\begin{array}{rcl} & \left ( \frac{\mathrm{d}} {\mathrm{d}z}\right )^{m}[z^{\beta -1}E_{ m/n,\beta }(z^{m/n})] & \\ & = z^{\beta -1}E_{m/n,\beta }(z^{m/n}) + z^{\beta -1}\,\sum _{k=1}^{n} \frac{z^{-\frac{m}{n}k}} {\varGamma \left (\beta -\frac{m} {n} k\right )}\quad (m,\ n \geq 1).&{}\end{array}$$
(4.3.2)
Since
$$\displaystyle{ \frac{1} {\varGamma (-s)} = 0\quad (s = 0,\ 1,\ 2,\ldots ),}$$
it follows from (4.3.2) with n = 1 and any \(\beta = 0,\ 1,\ldots,\ m\) that
$$\displaystyle{ \left ( \frac{\mathrm{d}} {\mathrm{d}z}\right )^{m}[z^{\beta -1}E_{ m,\beta }(z^{m})] = z^{\beta -1}E_{ m,\beta }(z^{m})\quad (m \geq 1). }$$
(4.3.3)
Substituting z nm in place of z in (4.3.2) we get
$$\displaystyle\begin{array}{rcl} & \left (\frac{m} {n} z^{1-\frac{n} {m} } \frac{\mathrm{d}} {\mathrm{d}z}\right )^{m}\left [z^{(\beta -1) \frac{n} {m} }E_{m,\beta }(z)\right ] & \\ & = z^{(\beta -1) \frac{n} {m} }E_{m,\beta }(z) + z^{(\beta -1) \frac{n} {m} }\,\sum _{k=1}^{n} \frac{z^{-k}} {\varGamma \left (\beta -\frac{m} {n} k\right )}\quad (m,\ n = 1,\ 2,\ldots ).&{}\end{array}$$
(4.3.4)
Let m = 1 in this formula. We then obtain the first-order differential equation for the function \(z^{(\beta -1)n}E_{1/n,\beta }(z)\):
$$\displaystyle\begin{array}{rcl} & \frac{1} {n} \frac{\mathrm{d}} {\mathrm{d}z}\left [z^{(\beta -1)n}E_{ 1/n,\beta }(z)\right ] - z^{n-1}[z^{(\beta -1)n}E_{ 1/n,\beta }(z)]& \\ & = z^{\beta n-1}\sum _{k=1}^{n} \frac{z^{-k}} {\varGamma \left (\beta -\frac{k} {n}\right )}\,,\quad (n = 1,\ 2,\ldots ). &{}\end{array}$$
(4.3.5)
Solving this equation we obtain for any z 0 ≠ 0 , 
$$\displaystyle\begin{array}{rcl} & E_{1/n,\beta }(z) = z^{(1-\beta )n}\,\mathrm{e}^{z^{n} }\left \{z_{0}^{(\beta -1)n}\,\mathrm{e}^{-z_{0}^{n} }\,E_{1/n,\beta }(z_{0})\right.& \\ & \left.+n\int _{z_{0}}^{z}\mathrm{e}^{-\tau ^{n} }\left (\sum _{k=1}^{n} \frac{\tau ^{-k}} {\varGamma \left (\beta -\frac{k} {n}\right )}\tau ^{\beta n-1}\right )\mathrm{d}\tau \right \}\quad (n = 1,\ 2,\ldots ). &{}\end{array}$$
(4.3.6)
Formula (4.3.6) is true with z 0 = 0 if β = 1. In this case we have
$$\displaystyle{ E_{1/n,1}(z) =\mathrm{ e}^{z^{n} }\left \{1 + n\int _{0}^{z}\mathrm{e}^{-\tau ^{n} }\,\left (\sum _{k=1}^{n-1}\frac{\tau ^{k-1}} {\varGamma \left (\frac{k} {n}\right )}\right )\mathrm{d}\tau \right \}\quad (n \geq 2). }$$
(4.3.7)
In particular,
$$\displaystyle{ E_{1/2,1}(z) =\mathrm{ e}^{z^{2} }\,\left \{1 + \frac{2} {\sqrt{\pi }}\int _{0}^{z}\mathrm{e}^{-\tau ^{2} }\mathrm{d}\tau \right \} =\mathrm{ e}^{z^{2} }\,\left \{1 + \mathrm{erf}\,z\right \} =\mathrm{ e}^{z^{2} }\,\mathrm{erfc}\,(-z)\,, }$$
(4.3.8)
and, consequently,
$$\displaystyle{E_{1/2,1}(z) \sim 2\,\mathrm{e}^{z^{2} },\quad \vert \mathrm{arg}z\vert < \frac{\pi } {4},\quad \vert z\vert \rightarrow \infty.}$$

In the following lemma we collect together a number of known recurrence relations for the two-parametric Mittag-Leffler function.

Lemma 4.1 ([GupDeb07]).

For all α > 0, β > 0 the following relations hold
$$\displaystyle\begin{array}{rcl} z^{2}E_{\alpha,\beta +2\alpha }(z) = E_{\alpha,\beta }(z) - \frac{1} {\varGamma (\beta )} - \frac{z} {\varGamma (\beta +\alpha )},& &{}\end{array}$$
(4.3.9)
$$\displaystyle\begin{array}{rcl} z^{3}E_{\alpha,\beta +3\alpha }(z) = E_{\alpha,\beta }(z) - \frac{1} {\varGamma (\beta )} - \frac{z} {\varGamma (\beta +\alpha )} - \frac{z^{2}} {\varGamma (\beta +2\alpha )},& &{}\end{array}$$
(4.3.10)
$$\displaystyle\begin{array}{rcl} z^{4}E_{\alpha,\beta +3\alpha }(z) = E_{\alpha,\beta }(z) - \frac{1} {\varGamma (\beta )} - \frac{z} {\varGamma (\beta +\alpha )} - \frac{z^{2}} {\varGamma (\beta +2\alpha )} - \frac{z^{3}} {\varGamma (\beta +3\alpha )}.& &{}\end{array}$$
(4.3.11)

4.4 Integral Relations and Asymptotics

Using the well-known discrete orthogonality relation
$$\displaystyle{\sum _{h=0}^{m-1}\mathrm{e}^{i2\pi \mathit{hk}/m} = \left \{\begin{array}{@{}l@{\quad }l@{}} m,\quad &\mathrm{if}\;\;\;k \equiv 0\pmod m\,, \\ 0, \quad &\mathrm{if}\;\;\;k\not\equiv 0\pmod m\, \end{array} \right.}$$
and definition (4.1.1) of the function E α, β (z) we have
$$\displaystyle{ \sum _{h=0}^{m-1}E_{\alpha,\beta }(z\,\mathrm{e}^{i2\pi h/m}) = m\,E_{\alpha /m,\beta }(z^{m})\quad (m \geq 1). }$$
(4.4.1)
Substituting here for α and z 1∕m for z we obtain
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {m}\sum _{h=0}^{m-1}E_{ m\alpha,\beta }(z^{1/m}\mathrm{e}^{i2\pi h/m})\quad (m \geq 1). }$$
(4.4.2)
Similarly, the formula
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {2m + 1}\sum _{h=-m}^{m}E_{ (2m+1)\alpha,\beta }(z^{1/(2m+1)}\mathrm{e}^{i2\pi h/(2m+1)})\quad (m \geq 0) }$$
(4.4.3)
can be obtained via the relation
$$\displaystyle{\sum _{h=-m}^{m}e\mathrm{e}^{i2\pi \mathit{hk}/(2m+1)} = \left \{\begin{array}{ll} 2m + 1,&\quad \mathrm{if}\;\;\;k \equiv 0\qquad \pmod 2m + 1\,, \\ 0, &\quad \mathrm{if}\;\;\;k\not\equiv 0\qquad \pmod 2m + 1\,. \end{array} \right.}$$
Using (4.1.1) and term-by-term integration we arrive at
$$\displaystyle{ \int _{0}^{z}E_{\alpha,\beta }(\lambda t^{\alpha })t^{\beta -1}\mathrm{d}t = z^{\beta }E_{\alpha,\beta +1}(\lambda z^{\alpha })\quad (\beta > 0), }$$
(4.4.4)
and furthermore, at the more general relation
$$\displaystyle\begin{array}{rcl} & \frac{1} {\varGamma (\alpha )}\int _{0}^{z}(z - t)^{\mu -1}\,E_{\alpha,\beta }(\lambda t^{\alpha })\,t^{\beta -1}\,\mathrm{d}t & \\ & = z^{\mu +\beta -1}E_{\alpha,\mu +\beta }(\lambda z^{\alpha })\quad (\mu > 0,\ \beta > 0),&{}\end{array}$$
(4.4.5)
where the integration is performed along the straight line connecting the points 0 and z.
It follows from formulas (4.4.5), (4.4.2) and (4.4.3) that
$$\displaystyle\begin{array}{rcl} \frac{1} {\varGamma (\beta )}\int _{0}^{z}(z - t)^{\beta -1}\mathrm{e}^{\lambda t}\mathrm{d}t = z^{\beta }E_{ 1,\beta +1}(\lambda z)\quad (\beta > 0),& &{}\end{array}$$
(4.4.6)
$$\displaystyle\begin{array}{rcl} \frac{1} {\varGamma (\beta )}\int _{0}^{z}(z - t)^{\beta -1}\cosh \sqrt{\lambda }t\,\mathrm{d}t = z^{\beta }E_{ 2,\beta +1}(\lambda z^{2})\quad (\beta > 0),& &{}\end{array}$$
(4.4.7)
$$\displaystyle\begin{array}{rcl} \frac{1} {\varGamma (\beta )}\int _{0}^{z}(z - t)^{\beta -1}\frac{\sinh \sqrt{\lambda }t} {\sqrt{\lambda }}\,\mathrm{d}t = z^{\beta +1}E_{ 2,\beta +2}(\lambda z^{2})\quad (\beta > 0).& &{}\end{array}$$
(4.4.8)
Let us prove the relation
$$\displaystyle\begin{array}{rcl} & z^{\beta -1}E_{\alpha,\beta }(z^{\alpha }) = z^{\beta -1}E_{2\alpha,\beta }(z^{2\alpha }) & \\ & + \frac{1} {\varGamma (\alpha )}\int _{0}^{z}(z - t)^{\alpha -1}E_{ 2\alpha,\beta }(t^{2\alpha })t^{\beta -1}\,\mathrm{d}t\quad (\beta > 0).&{}\end{array}$$
(4.4.9)
First of all, we have by direct evaluations
$$\displaystyle\begin{array}{rcl} & \int _{0}^{z}E_{2\alpha,\beta }(t^{2\alpha })t^{\beta -1}\left \{1 + \frac{(z-t)^{\alpha }} {\varGamma (\alpha +1)} \right \}\,\mathrm{d}t & {}\\ & =\sum _{ k=0}^{\infty } \frac{1} {\varGamma (2k\alpha +\beta )}\int _{0}^{z}t^{2k\alpha +\beta -1}\left \{1 + \frac{(z-t)^{\alpha }} {\varGamma (\alpha +1)} \right \}\,\mathrm{d}t & {}\\ & = z^{\beta }\sum _{k=0}^{\infty } \frac{z^{2k\alpha }} {\varGamma (2k\alpha +\beta +1)} + z^{\beta }\sum _{ k=0}^{\infty } \frac{z^{(2k+1)\alpha }} {\varGamma ((2k+1)\alpha +\beta +1)}& {}\\ & = z^{\beta }\sum _{k=0}^{\infty } \frac{z^{k\alpha }} {\varGamma (k\alpha +\beta +1)} = z^{\beta }E_{\alpha,\beta +1}(z^{\alpha }). & {}\\ \end{array}$$
This relation and formula (4.4.4) imply
$$\displaystyle\begin{array}{rcl} & \int _{0}^{z}E_{2\alpha,\beta }(t^{2\alpha })t^{\beta -1}\left \{1 + \frac{(z-t)^{\alpha }} {\varGamma (\alpha +1)} \right \}\,\mathrm{d}t& {}\\ & =\int _{ 0}^{z}E_{\alpha,\beta }(t^{\alpha })t^{\beta -1}\,\mathrm{d}t\quad (\beta > 0). & {}\\ \end{array}$$
Differentiation of this formula with respect to z gives us formula (4.4.9).
Let us prove the formula
$$\displaystyle\begin{array}{rcl} & \int _{0}^{l}x^{\beta -1}E_{\alpha,\beta }(\lambda x^{\alpha })\,(l - x)^{\nu -1}E_{\alpha,\nu }(\lambda ^{{\ast}}(l - x)^{\alpha })\,\mathrm{d}x& \\ & = \frac{\lambda \,E_{\alpha,\beta +\nu }(l^{\alpha }\lambda )-\lambda ^{{\ast}}\,E_{\alpha,\beta +\nu }(l^{\alpha }\lambda ^{{\ast}})} {\lambda -\lambda ^{{\ast}}} \,l^{\beta +\nu -1}\quad (\beta > 0\,,\;\nu > 0)\,, &{}\end{array}$$
(4.4.10)
where λ and \(\lambda ^{{\ast}}\ (\lambda \not =\lambda ^{{\ast}})\) are any complex parameters.
Indeed, using (4.1.1) for any λ and λ (λ ≠ λ ) and β > 0 ,  ν > 0 we find
$$\displaystyle\begin{array}{rcl} & \int _{0}^{l}x^{\beta -1}E_{\alpha,\beta }(\lambda x^{\alpha })(l - x)^{\nu -1}E_{\alpha,\nu }(\lambda ^{{\ast}}(l - x)^{\alpha })\,\mathrm{d}x & {}\\ & =\sum _{ n=0}^{\infty }\sum _{m=0}^{\infty } \frac{\lambda ^{n}(\lambda ^{{\ast}})^{m}} {\varGamma (n\alpha +\beta )\,\varGamma (m\alpha +\nu )}\int _{0}^{l}x^{n\alpha +\beta -1}(l - x)^{m\alpha +\nu -1}\,\mathrm{d}x & {}\\ & =\sum _{ n=0}^{\infty }\sum _{m=0}^{\infty }\frac{\lambda ^{n}(\lambda ^{{\ast}})^{m}l^{(n+m)\alpha +\beta +\nu -1}} {\varGamma ((m+n)\alpha )+\beta +\nu )} = l^{\beta +\nu -1}\sum _{ n=0}^{\infty }\sum _{ k=n}^{\infty }\frac{\lambda ^{n}(\lambda ^{{\ast}})^{k-n}\,l^{k\alpha }} {\varGamma (k\alpha +\beta +\nu )} & {}\\ & = l^{\beta +\nu -1}\sum _{k=0}^{\infty }\frac{(\lambda ^{{\ast}})^{k}\,l^{k\alpha }} {\varGamma (k\alpha +\beta +\nu )}\,\sum _{n=0}^{k}\left ( \frac{\lambda }{\lambda ^{ {\ast}}}\right )^{n} = \frac{l^{\beta +\nu -1}} {\lambda -\lambda ^{{\ast}}} \sum _{k=0}^{\infty }\frac{l^{k\alpha }(\lambda ^{k+1}-(\lambda ^{{\ast}})^{k+1})} {\varGamma (k\alpha +\beta +\nu )} \,. & {}\\ \end{array}$$
Using formula (4.1.1) once more, we arrive at (4.4.10).
Finally, we obtain two integral relations:
$$\displaystyle\begin{array}{rcl} \int _{0}^{+\infty }\mathrm{e}^{-t}E_{\alpha,\beta }(\mathit{zt}^{\alpha })t^{\beta -1}\mathrm{d}t = \frac{1} {1 - z}\quad (\beta > 0,\ \vert z\vert < 1)\,,& &{}\end{array}$$
(4.4.11)
$$\displaystyle\begin{array}{rcl} \int _{0}^{+\infty }\mathrm{e}^{-t^{2}/(4x) }\,E_{\alpha,\beta }(t^{\alpha })\,t^{\beta -1}\,\mathrm{d}t = \sqrt{\pi }x^{\beta /2}\,E_{ 2\alpha,\frac{1+\beta } {2} }\left (x^{2\alpha }\right )\;(\beta > 0,\ x > 0).& &{}\end{array}$$
(4.4.12)
First of all, since the Mittag-Leffler type function (4.1.1) is an entire function of order ρ = 1∕(Re α) and type 1 (see Sect. 4.1), we have for any σ > 1 the estimate:
$$\displaystyle{\vert E_{\alpha,\beta }(z)\vert \leq C\mbox{ exp}\{\sigma \vert z\vert ^{\rho }\}.}$$
Consequently, the integrals in the formulae (4.4.11) and (4.4.12) are convergent.
It is easy to check that term-by-term integration of the expansion
$$\displaystyle{\mathrm{e}^{-t}\,E_{\alpha,\beta )}(\mathit{zt}^{\alpha })t^{\beta -1} =\sum _{ k=0}^{\infty }\frac{\mathrm{e}^{-t}\,t^{k\alpha +\beta -1}} {\varGamma (k\alpha +\beta )} \,z^{k}}$$
can be performed with respect to t along the interval (0,  + ) if | z |  < 1. As a result, we arrive at formula (4.4.11).
Similarly, we have using term-by-term integration of the following expansion with respect to t:
$$\displaystyle{\mathrm{e}^{-t^{2}/(4x) }\,E_{\alpha,\beta }(t^{\alpha })\,t^{\beta -1} =\sum _{ k=0}^{\infty }\frac{t^{k\alpha +\beta -1}} {\varGamma (k\alpha +\beta )} \,\mathrm{e}^{-t^{2}/(4x) }\quad (\beta > 0)}$$
along the same interval (0,  + ) (integration is performable with any fixed x > 0!),
$$\displaystyle{\int _{0}^{+\infty }\,\mathrm{e}^{-t^{2}/(4x) }\,E_{\alpha,\beta }(t^{\alpha })\,t^{\beta -1}\,\mathrm{d}t =\sum _{ k=0}^{\infty } \frac{\varGamma \left (\frac{k\alpha +\beta } {2} \right )} {2\,\varGamma (k\alpha +\beta )}\,(2\sqrt{x})^{k\alpha +\beta }\,.}$$
Rewriting the right-hand side of the last relation by using the Lagrange formula
$$\displaystyle{\varGamma (s)\varGamma (s + 1/2) = \sqrt{\pi }\,2^{1-2s}\,\varGamma (2s)}$$
we obtain formula (4.4.12).

The Mittag-Leffler type function \(E_{\alpha,\alpha }\left (z\right )\) plays an essential role in the linear Abel integral equation of the second kind (see, e.g., [GorVes91, GorMai97]).

Theorem 4.2.

Let a function f(t) be in the function space L 1 (0, l). Let α > 0 and λ be an arbitrary complex parameter. Then the integral equation
$$\displaystyle{ u(t) = f(t) + \frac{\lambda } {\varGamma (\alpha )}\int _{0}^{t}(t-\tau )^{\alpha -1}\,u(\tau )\,\mathrm{d}\tau,\quad t \in (0,\ l), }$$
(4.4.13)
has a unique solution
$$\displaystyle{ u(t) = f(t) +\lambda \int _{ 0}^{t}(t-\tau )^{\alpha -1}\,E_{\alpha,\alpha }\left [\lambda \,(t-\tau )^{\alpha }\right )\,f(\tau )\,\mathrm{d}\tau,\quad t \in (0,\ l), }$$
(4.4.14)
in the space L 1 (0, l) .

This result was discovered in the pioneering work of Hille and Tamarkin [HilTam30]. We give the proof of this theorem in Chap.  7.

We consider now a simple application of Theorem 4.2.

Let z = t > 0 and t = τ be the integration variable in (4.4.9). Then this representation can be considered as an integral equation of type (4.4.13) with
$$\displaystyle{f(t) = t^{\beta -1}\,E_{\alpha,\beta }(t^{\alpha })\,,\quad \lambda = 1\,;}$$
and solution
$$\displaystyle{u(t) = t^{\beta -1}E_{ 2\alpha,\beta }(x^{2\alpha })\,.}$$
Using formula (4.4.14) we obtain for the solution u(t) the representation
$$\displaystyle\begin{array}{rcl} & u(t) = t^{\beta -1}\,E_{\alpha,\beta }(t^{\alpha }) & \\ & -\int _{0}^{t}(t-\tau )^{\alpha -1}\,E_{\alpha,\beta }(\tau ^{\alpha })\,\tau ^{\beta -1}\,E_{\alpha,\alpha }\left [-(t-\tau )^{\alpha }\right ]\mathrm{d}\tau.&{}\end{array}$$
(4.4.15)

We also mention asymptotic results for the Mittag-Leffler function E α, β (z) which are essentially a refinement of results of Dzhrbashyan (see [Dzh66]).

Theorem 4.3.

For all 0 < α < 2, \(\beta \in \mathbb{C}\) , \(m \in \mathbb{N}\) , the following asymptotic formulas hold:

If |arg  z| < min {π,πα}, then
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } } -\sum \limits _{k=1}^{m} \frac{z^{-k}} {\varGamma (\beta -k\alpha )} + O\left (\vert z\vert ^{-m-1}\right ),\,\vert z\vert \rightarrow \infty. }$$
(4.4.16)
If 0 < α < 1, πα < |arg  z| < π, then
$$\displaystyle{ E_{\alpha,\beta }(z) = -\sum \limits _{k=1}^{m} \frac{z^{-k}} {\varGamma (\beta -k\alpha )} + O\left (\vert z\vert ^{-m-1}\right ),\,\vert z\vert \rightarrow \infty. }$$
(4.4.17)

Theorem 4.4.

For all α ≥ 2, \(\beta \in \mathbb{C}\) , \(m \in \mathbb{N}\) , the following asymptotic formula holds:
$$\displaystyle\begin{array}{rcl} & E_{\alpha,\beta }(z) = \frac{1} {\alpha } \sum \limits _{\vert \mathrm{arg}\,z+2\pi n\vert <\frac{3\pi \alpha } {4} }\left (z^{1/\alpha }\mathrm{e}^{2\pi \mathit{in}/\alpha }\right )^{1-\beta }\mathrm{e}^{z^{1/\alpha }\mathrm{e}^{2\pi \mathit{in}/\alpha } }& \\ & -\sum \limits _{k=1}^{m} \frac{z^{-k}} {\varGamma (\beta -k\alpha )} + O\left (\vert z\vert ^{-m-1}\right ),\,\vert z\vert \rightarrow \infty. &{}\end{array}$$
(4.4.18)

A more exact description of the remainders in formulas (4.4.16)–(4.4.18) is presented, e.g., in [PopSed11].

4.5 The Two-Parametric Mittag-Leffler Function as an Entire Function

One more immediate consequence of the results in [Sed94, Sed00, Sed04, Sed07] (see also [Pop02, Pop06, PopSed03]) is the fact that E α, β with \(\alpha > 0,\beta \in \mathbb{R}\), is a function of completely regular growth in the sense of Levin–Pfluger.

For an entire function F(z) of finite order ρ this means that the following limit exists
$$\displaystyle{ h_{F}(\theta ) =\lim _{r\rightarrow \infty }\frac{\log \,\left \vert F(r\mathrm{e}^{i\theta })\right \vert } {r^{\rho }}, }$$
(4.5.1)
when r →  takes all positive values avoiding an exceptional set C 0 of relatively small measure common for all rays arg z = θ (see the formal definition in Sect. B.4). Sometimes the limit in (4.5.1) is called the weak limit and is denoted by
$$\displaystyle{\lim _{r\rightarrow \infty }^{\!\!\!{\ast}}\,.}$$
The above property is known to be equivalent to the regularity of the distribution of zeros of an entire function (see [Lev56, Ron92]). The corresponding result is presented in Sect. B.4.
To prove that E α, β possesses this property let us consider the Weierstrass product representation (see Theorem B.3.1). For the two-parametric Mittag-Leffler function this representation has the form
$$\displaystyle\begin{array}{rcl} & E_{\alpha,\beta } =\prod \limits _{ n=1}^{\infty }\left (1 - \frac{z} {z_{n}}\right )\mathrm{e}^{ \frac{z} {z_{n}} + \frac{z^{2}} {2z_{n}^{2}} +\ldots + \frac{z^{[1/\alpha ]}} {[1/\alpha ]z_{n}^{[1/\alpha ]}} } & \\ & \times \prod \limits _{n=1}^{\infty }\left (1 - \frac{z} {z_{-n}}\right )\mathrm{e}^{ \frac{z} {z_{-n}} + \frac{z^{2}} {2z_{-n}^{2}} +\ldots + \frac{z^{[1/\alpha ]}} {[1/\alpha ]z_{-n}^{[1/\alpha ]}} } \times \mathrm{ e}^{q_{0}+q_{1}z+\ldots +q_{[1/\alpha ]}z^{[1/\alpha ]} },&{}\end{array}$$
(4.5.2)
where z n , z n are the zeros of E α, β in the neighbourhoods of the rays \(\mathrm{arg}\,z = \frac{\pi \alpha } {2},\;\mathrm{arg}\,z = -\frac{\pi \alpha }{2}\), respectively.
This formula can be written in terms of simple Weierstrass factors:
$$\displaystyle{ G(\zeta,p) = (1-\zeta )\mathrm{e}^{\zeta +\frac{\zeta ^{2}} {2} +\ldots +\frac{\zeta ^{p}} {p} },\;\;\;p \in \mathbb{N}. }$$
(4.5.3)
Then (4.5.2) becomes
$$\displaystyle{ E_{\alpha,\beta }(z) =\prod \limits _{ n=1}^{\infty }G\left ( \frac{z} {z_{n}},[1/\alpha ]\right ) \times \prod \limits _{n=1}^{\infty }G\left ( \frac{z} {z_{-n}},[1/\alpha ]\right ) \times \mathrm{ e}^{q_{0}+q_{1}z+\ldots +q_{[1/\alpha ]}z^{[1/\alpha ]} }. }$$
(4.5.4)
Formula (4.5.4) (or (4.5.2)) is not too useful for obtaining asymptotic results. By using the technique developed for the entire functions of completely regular growth (see, e.g., [GoLeOs91, Lev56, Ron92]), we can compare the asymptotic behaviour of E α, β with that of more simple functions. In order to formulate it in a more rigorous form let us introduce two pairs of sequences:
$$\displaystyle\begin{array}{rcl} w_{n} = \vert z_{n}\vert \mathrm{e}^{i \frac{\pi \alpha }{2} },\;n = 1,2,,\ldots,\;w_{-n} = \vert z_{-n}\vert \mathrm{e}^{-i \frac{\pi \alpha }{2} },\;n = 1,2,,\ldots;& &{}\end{array}$$
(4.5.5)
$$\displaystyle\begin{array}{rcl} \omega _{n} = (2\pi n)^{\alpha }\mathrm{e}^{i \frac{\pi \alpha }{2} },\;n = 1,2,,\ldots,\;\omega _{-n} = (2\pi n)^{\alpha }\mathrm{e}^{-i \frac{\pi \alpha }{2} },\;n = 1,2,,\ldots;& &{}\end{array}$$
(4.5.6)
and construct the corresponding Weierstrass products for these sequences
$$\displaystyle\begin{array}{rcl} W(z) =\prod \limits _{ n=1}^{\infty }G\left ( \frac{z} {w_{n}},[1/\alpha ]\right ) \times \prod \limits _{n=1}^{\infty }G\left ( \frac{z} {w_{-n}},[1/\alpha ]\right ) \times \mathrm{ e}^{q_{0}+q_{1}z+\ldots +q_{[1/\alpha ]}z^{[1/\alpha ]} },& &{}\end{array}$$
(4.5.7)
$$\displaystyle\begin{array}{rcl} \varOmega (z) =\prod \limits _{ n=1}^{\infty }G\left (\frac{z} {\omega _{n}},[1/\alpha ]\right ) \times \prod \limits _{n=1}^{\infty }G\left ( \frac{z} {\omega _{-n}},[1/\alpha ]\right ) \times \mathrm{ e}^{q_{0}+q_{1}z+\ldots +q_{[1/\alpha ]}z^{[1/\alpha ]} }.& &{}\end{array}$$
(4.5.8)

Lemma 4.5.

The functions E α,β (z), W(z), Ω(z) are functions of completely regular growth with the same characteristics, i.e. with the same angular density (see formula (B.4.1a) ) and in the case of integer ρ = [1∕α] also with the same coefficients of angular symmetry (see formula (B.4.1b)) .

To prove this result it suffices to examine the corresponding properties of the sequences \((z_{\pm n}),(w_{\pm n}),(\omega _{\pm n})\).

Lemma 4.6.

The functions E α,β (z), W(z), Ω(z) have the same asymptotic behaviour, i.e. the following weak limits exist 1:
$$\displaystyle{ \lim _{r\rightarrow \infty }^{\!\!\!{\ast}}\frac{\left \vert E_{\alpha,\beta }(r\mathrm{e}^{i\theta })\right \vert } {\left \vert W(r\mathrm{e}^{i\theta })\right \vert } = \lim _{r\rightarrow \infty }^{\!\!\!{\ast}}\frac{\left \vert W(r\mathrm{e}^{i\theta })\right \vert } {\left \vert \varOmega (r\mathrm{e}^{i\theta })\right \vert } = 1. }$$
(4.5.9)

\(\vartriangleleft \) Comparing representations (4.5.4) and (4.5.7) we arrive at the conclusion that the quotient under this limit is in fact a Blaschke type product for two rays. Such products were studied in [Gov94]. Repeating calculations of [Gov94] we get the existence of the first limit (4.5.9).

The second result of the lemma also follows from quite general considerations. Comparing sequences (w ±n ) and (ω ±n ) we can see that there exists a proximate order (cf. [Lev56, GoLeOs91]) (i.e. a non-negative, non-decreasing function such that \(\lim _{r\rightarrow \infty }\rho (r) =\rho\), \(\lim _{r\rightarrow \infty }\rho ^{{\prime}}(r)r\log \,r = 0\)) for which
$$\displaystyle{ \left \vert w_{\pm n}\right \vert ^{\rho (r)} = \left \vert \omega _{ \pm n}\right \vert ^{\rho }. }$$
(4.5.10)
A possible way to see this is to use an interpolating \(\mathcal{C}^{1}\)-function ρ(r) obeying the interpolation conditions
$$\displaystyle{ \rho (n) =\rho \frac{\log \,\vert \omega _{\pm n}\vert } {\log \,\vert w_{\pm n}\vert },\;\;\;n = 1,2,\ldots }$$
(4.5.11)
Since we need to obtain only an asymptotic result, the interpolation problem can be simplified and even solved in closed form. After constructing the proximate order it remains to apply the Valiron–Levin theory of entire functions of a proximate order (see, e.g., [Lev56]). This completes the proof.\(\vartriangleright \)
We are now in a position to discuss the global behaviour of the Mittag-Leffler type function E α, β (z). This behaviour is described in terms of the indicator function for E α, β (z) in the case 0 < α < 2 (we assume additionally that β ≠ 1, 0, −1, , when α = 1, see (B.2.5)):
$$\displaystyle{ h_{E_{\alpha,\beta }}(\theta ) = \left \{\begin{array}{ll} \cos \,\frac{\theta }{\alpha }, &0 \leq \vert \theta \vert < \frac{\pi \alpha } {2}, \\ 0,& \frac{\pi \alpha } {2} \leq \vert \theta \vert \leq \pi. \end{array} \right.}$$
(4.5.12)

4.6 Distribution of Zeros

In this section we mainly follow the article [Sed94] (see also [PopSed11]).

First of all we recall some relations of the Mittag-Leffler function to elementary functions for special values of the parameters α, β. In these cases the zeros of the Mittag-Leffler function can be found explicitly.
$$\displaystyle\begin{array}{rcl} E_{1,1}(z) = \mbox{ exp}\{z\},\;E_{1,-m}(z) = z^{m+1}\mbox{ exp}\{z\},\;m \in \mathbb{Z}_{ +};& &{}\end{array}$$
(4.6.1)
$$\displaystyle\begin{array}{rcl} E_{2,1}(z) =\cosh \sqrt{z},\;E_{2,2}(z) = \frac{\sinh \sqrt{z}} {\sqrt{z}},\;E_{2,3}(z) = \frac{\cosh \sqrt{z} - 1} {\sqrt{z}},& &{}\end{array}$$
(4.6.2)
$$\displaystyle\begin{array}{rcl} E_{2,-2m}(z) = z^{m+1/2}\sinh \sqrt{z},\;m \in \mathbb{Z}_{ +},\;E_{2,-(2m-1)}(z) = z^{m}\cosh \sqrt{z},\;m \in \mathbb{N}.& &{}\end{array}$$
(4.6.3)
Thus the function E 1, 1(z) has no zero, while the function E 1, −m (z) has its only zero at z = 0 of order m + 1. Below we can see that for all other values of parameters α, β (including those described in (4.6.2) and (4.6.3)) the Mittag-Leffler function E α, β (z) has an infinite number of zeros. We should also mention that the function E 2, 3(z) has double zeros at the points \(z_{n} = -(2\pi n)^{2},\,n \in \mathbb{N}\). This is the only case when the Mittag-Leffler function E α, β (z) has an infinite number of multiple zeros.
In order to describe the distribution of zeros of the Mittag-Leffler function E α, β (z) we introduce the following constants:
$$\displaystyle{ c_{\beta } = \frac{\alpha } {\varGamma (\beta -\alpha )},\;d_{\beta } = \frac{\alpha } {\varGamma (\beta -2\alpha )},\;\tau _{\beta } = 1 + \frac{1-\beta } {\alpha },\;\beta \not =\alpha - l,\,l \in \mathbb{Z}_{+}, }$$
(4.6.4)
$$\displaystyle{ c_{\beta } = \frac{\alpha } {\varGamma (\beta -2\alpha )},\;d_{\beta } = \frac{\alpha } {\varGamma (\beta -3\alpha )},\;\tau _{\beta } = 2 + \frac{1-\beta } {\alpha },\;\beta =\alpha -l,\,l \in \mathbb{Z}_{+},\alpha \not\in \mathbb{N}. }$$
(4.6.5)
By construction c β  ≠ 0 for all considered α and β. The values of parameters α and β, which are not mentioned in (4.6.4) and (4.6.5), are called exceptional values.

Theorem 4.7.

Let the values of parameters α,β satisfy one of the following relations:
  1. (1)

    \(0 <\alpha < 2,\beta \in \mathbb{C}\) ,  and β≠0,−1,−2,… when α = 1;

     
  2. (2)

    α = 2, Re  β > 3.

     
Then all zeros z n of the Mittag-Leffler function E α,β (z) with sufficiently large modulus are simple and the asymptotic relation holds for n →±∞
$$\displaystyle{ \left (z_{n}\right )^{1/\alpha } = 2\pi \mathit{in} -\tau _{\beta }\alpha \log \,2\pi \mathit{in} + \frac{d_{\beta }/c_{\beta }} {(2\pi \mathit{in})^{\alpha }} + (\tau _{\beta }\alpha )^{2}\frac{\log \,2\pi \mathit{in}} {2\pi \mathit{in}} - (\tau _{\beta }\alpha )^{2} \frac{\log \,c_{\beta }} {2\pi \mathit{in}} +\alpha _{n}, }$$
(4.6.6)
where for α < 2
$$\displaystyle{ \alpha _{n} = O\left ( \frac{\log \vert n\vert } {\vert n\vert ^{1+\alpha }}\right ) + O\left ( \frac{1} {\vert n\vert ^{2\alpha }}\right ) + O\left (\frac{\log ^{2}\vert n\vert } {\vert n\vert ^{2}}\right ), }$$
(4.6.7)
but for α = 2
$$\displaystyle{ \alpha _{n} = \frac{\mathrm{e}^{\pm i\pi \beta }} {c_{\beta }^{2}(2\pi n)^{-4\tau _{\beta }}} + O\left ( \frac{1} {\vert n\vert ^{-8\tau _{\beta }}}\right ) + O\left ( \frac{\log \vert n\vert } {\vert n\vert ^{1-4\tau _{\beta }}}\right ) + O\left (\frac{\log ^{2}\vert n\vert } {\vert n\vert ^{2}}\right ). }$$
(4.6.8)

The proof of the theorem is based on the integral representation of the Mittag-Leffler function E α, β (z) and on the following lemma.

Lemma 4.8.

Let \(A \in \mathbb{C}\) and δ ∈ (0,π∕2) be fixed numbers and let the sets Z, W be defined for all R > 0 by the relations
$$\displaystyle\begin{array}{rcl} & Z = Z_{\delta,R} = \left \{z: \vert \mathrm{arg}\,z\vert <\pi -\delta,\,\vert z\vert > R\right \}, & {}\\ & W = W_{\delta,R} = \left \{z: \vert \mathrm{arg}\,z\vert <\pi -2\delta,\,\vert z\vert > 2R\right \}.& {}\\ \end{array}$$
Then for sufficiently large R > 0 the equation
$$\displaystyle{z - A\log \,z = w,\;w \in W,}$$
has a unique zero z ∈ Z. This zero is simple and we have the asymptotics
$$\displaystyle{ z = w + A\log \,w + A^{2}\frac{\log \,w} {w} + O\left (\frac{\log ^{2}\,w} {w^{2}}\right ),\;w \rightarrow \infty. }$$
(4.6.9)
\(\vartriangleleft \) Proof of Theorem 4.7 ([PopSed11, pp. 35–37]). Let us put in the formulas (4.4.16) and (4.4.17) m = 1 if β is defined as in (4.6.4), and m = 2 if β is defined as in (4.6.5). Then c β  ≠ 0. Hence one can conclude from (4.4.16) and (4.4.17) that for any \(\varepsilon > 0\) all zeros z n of the Mittag-Leffler function E α, β (z) with sufficiently large modulus are situated in the angle \(\vert \mathrm{arg}\,z\vert < \frac{\pi \alpha } {2}+\varepsilon\). In this angle E α, β (z) satisfies the asymptotic relation
$$\displaystyle{ \alpha z^{m}E_{\alpha,\beta }(z) = z^{\tau _{\beta }}\mbox{ exp}\{z^{1/\alpha }\} - c_{\beta } -\frac{d_{\beta }} {z} + O\left ( \frac{1} {z^{2}}\right ),\;\vert \mathrm{arg}\,z\vert < \frac{\pi \alpha } {2} +\varepsilon. }$$
(4.6.10)
Therefore, there exists a sufficiently large r 0 such that all zeros \(z_{n},\vert z_{n}\vert > r_{0}\), can be found from the equation
$$\displaystyle{ \mbox{ exp}\{z^{1/\alpha } +\tau _{\beta }\log \,z\} = c_{\beta } + \frac{d_{\beta }} {z} + O\left ( \frac{1} {z^{2}}\right ). }$$
(4.6.11)
Let us put
$$\displaystyle{ w = z^{1/\alpha } +\alpha \tau _{\beta }\log \,z^{1/\alpha }. }$$
(4.6.12)
Then, by Lemma 4.8 we obtain the solution to (4.6.12) with respect to z 1∕α in the form
$$\displaystyle{z^{1/\alpha } = w + O\left (\log \,w\right ).}$$
Hence
$$\displaystyle{\frac{1} {z} = \frac{1} {w^{\alpha }}\left (1 + O\left (\frac{\log \,w} {w}\right )\right ) = \frac{1} {w^{\alpha }} + O\left ( \frac{\log \,w} {w^{1+\alpha }}\right ).}$$
Substituting this relation into (4.6.11) we obtain the equation
$$\displaystyle{ \mbox{ exp}\{w\} = c_{\beta } + \frac{d_{\beta }} {w^{\alpha }} + O\left ( \frac{\log \,w} {w^{1+\alpha }}\right ) + O\left ( \frac{1} {w^{2\alpha }}\right ). }$$
(4.6.13)
In particular,
$$\displaystyle{ \mbox{ exp}\{w\} = c_{\beta } + o\left (1\right ),\;w \rightarrow \infty. }$$
(4.6.14)
Since all zeros of the function exp{w} − c β are simple and are given by the formula \(2\pi \mathit{in} +\log \, c_{\beta },\,n \in \mathbb{Z}\), by Rouché’s theorem all zeros w n of Eq. (4.6.14) with sufficiently large modulus are simple too and can be described by the formula
$$\displaystyle{ w_{n} = 2\pi \mathit{in} +\log \, c_{\beta } +\epsilon _{n},\;\epsilon _{n} \rightarrow 0,\;n \rightarrow \pm \infty. }$$
(4.6.15)
Thus
$$\displaystyle{ \frac{1} {w_{n}} = \frac{1} {2\pi \mathit{in}} + \left (1 + O\left ( \frac{1} {n}\right )\right ),\;\log \,w_{n} =\log \, \vert n\vert + O\left (1\right ),\;n \rightarrow \pm \infty. }$$
(4.6.16)
Therefore, if w = w n in (4.6.14), then
$$\displaystyle{c_{\beta }\mbox{ exp}\{\epsilon _{n}\} = c_{\beta } + \frac{d_{\beta }} {(2\pi \mathit{in})^{\alpha }} + O\left ( \frac{\log \,\vert n\vert } {\vert n\vert ^{1+\alpha }}\right ) + O\left ( \frac{1} {\vert n\vert ^{2\alpha }}\right ).}$$
Since the left-hand side of this relation is equal to \(c_{\beta } + c_{\beta }\epsilon _{n} + O\left (\epsilon _{n}^{2}\right )\), we have \(\epsilon _{n} = O\left ( \frac{1} {\vert n\vert ^{\alpha }}\right )\) and hence
$$\displaystyle{ \epsilon _{n} = \frac{d_{\beta }/c_{\beta }} {(2\pi \mathit{in})^{\alpha }} + O\left ( \frac{\log \,\vert n\vert } {\vert n\vert ^{1+\alpha }}\right ) + O\left ( \frac{1} {\vert n\vert ^{2\alpha }}\right ). }$$
(4.6.17)
Substituting this relation into (4.6.15) we get
$$\displaystyle\begin{array}{rcl} w_{n} = 2\pi \mathit{in} +\log \, c_{\beta } + \frac{d_{\beta }/c_{\beta }} {(2\pi \mathit{in})^{\alpha }} + O\left ( \frac{\log \,\vert n\vert } {\vert n\vert ^{1+\alpha }}\right ) + O\left ( \frac{1} {\vert n\vert ^{2\alpha }}\right ),\;n \rightarrow \pm \infty,& &{}\end{array}$$
(4.6.18)
$$\displaystyle\begin{array}{rcl} \log \,w_{n} =\log \, 2\pi \mathit{in} + \frac{\log \,c_{\beta }} {2\pi \mathit{in}} + O\left ( \frac{1} {\vert n\vert ^{1+\alpha }}\right ),\;n \rightarrow \pm \infty.& &{}\end{array}$$
(4.6.19)
Now we note that the pre-images z n of w n satisfy \(\vert \mathrm{arg}\,z_{n}\vert < \frac{\pi \alpha } {2}+\varepsilon\) and \(\frac{1} {\alpha } \left ( \frac{\pi \alpha } {2}+\varepsilon \right ) <\pi\). Thus the conditions of Lemma 4.8 are satisfied and we obtain from this lemma and from (4.6.12) and (4.6.18)
$$\displaystyle{ z_{n}^{1/\alpha } = w_{ n} -\alpha \tau _{\beta }\log \,w_{n} + \left (\alpha \tau _{\beta }\right )^{2}\frac{\log \,w_{n}} {w_{n}} + O\left (\frac{\log ^{2}\,w_{n}} {w_{n}^{2}}\right ). }$$
(4.6.20)
Then the proof of the theorem in case (1) follows from (4.6.18) and (4.6.19).
In case (2) one can use a similar argument (see [PopSed11, pp. 36–37]) based on the following asymptotic formula
$$\displaystyle{ E_{2,\beta }(z) = \frac{1} {2}z^{(1-\beta )/2}\left (\mathrm{e}^{\sqrt{z}} +\mathrm{ e}^{\mp i\pi (1-\beta )}\mathrm{e}^{-\sqrt{z}}\right ) -\sum \limits _{ k=1}^{m} \frac{1} {z^{k}\varGamma (\beta -2k)} + O\left ( \frac{1} {z^{m+1}}\right ), }$$
(4.6.21)
which is valid for | z | →  in the angles 0 ≤ arg z ≤ π and −π ≤ arg z ≤ 0, respectively. \(\vartriangleright \)

The most attractive result (see, e.g., [PopSed11, p. 37]) concerning the distribution of zeros of the Mittag-Leffler function is the following:

Theorem 4.9.

Let \(\mathrm{Re}\,\beta < 3,\beta \not =2 - l,l \in \mathbb{Z}_{+}\) . Then all zeros z n of the Mittag-Leffler function E 2,β with sufficiently large modulus are simple and the following asymptotic formula is valid (n →±∞)
$$\displaystyle{ \sqrt{z_{n}} =\pi i\left (n - 1 +\beta /2\right ) + (-1)^{n}\frac{c_{\beta }\mathrm{e}^{-i\pi \beta /2}} {2(i\pi n)^{2\tau _{\beta }}} + O\left ( \frac{1} {n^{6\mathrm{Re}\,\tau _{\beta }}}\right ) + O\left ( \frac{1} {n^{1+2\mathrm{Re}\,\tau _{\beta }}}\right ), }$$
(4.6.22)
where the single-valued branch of the function \(\sqrt{z}\) is chosen by the relation 0 ≤arg  z < 2π.

If β is real then all zeros z n of E 2,β with sufficiently large modulus are real.

Now we present a result on the distribution of zeros of the function E 2, 3+ (z), γ ≠ 0, \(\gamma \in \mathbb{R}\). For the proof of the following theorem we refer to [PopSed11, pp. 39–44].

Theorem 4.10.

  1. (1)

    The set of multiple zeros of the function \(E_{2,3+i\gamma }(z)\) is at most finite.

     
  2. (2)
    The sequence of all zeros z n consists of two subsequences \(z_{n}^{+},n > n^{+}\) , and \(z_{n}^{-},n < -n^{-}\) , for which the following asymptotic relation is valid
    $$\displaystyle{ \sqrt{z_{n }^{\pm }} = 2\pi \mathit{in} - \frac{\pi \gamma } {2} +\delta _{n} + O\left ( \frac{1} {n}\right ),\;n \rightarrow \pm \infty, }$$
    (4.6.23)
    where the sequence δ n is defined by
    $$\displaystyle{ \delta _{n} =\delta _{n}(\gamma ) =\log \, \left (\eta \mathrm{e}^{i\gamma \log \,2\pi n} + \sqrt{\left (\eta \mathrm{e}^{i\gamma \log \,2\pi n } \right ) ^{2 } - 1}\right ),\;\eta = \frac{1} {\varGamma (1 + i\gamma )}, }$$
    (4.6.24)
    where the principal branch is used for the values of the logarithmic function.
     
  3. (3)
    The sequence \(\zeta _{n} = \sqrt{z_{n}}\) asymptotically belongs to the semi-strips
    $$\displaystyle{ \log \,\rho _{1} < \left \vert \mathrm{Re}\,\zeta + \frac{\pi \gamma } {2}\right \vert <\log \,\rho _{2},\;\mathrm{Im}\,\zeta > 0, }$$
    (4.6.25)
    where
    $$\displaystyle{\rho _{1} = \vert \eta \vert + \sqrt{\vert \eta \vert ^{2 } - 1},\;\rho _{2} = \vert \eta \vert + \sqrt{\vert \eta \vert ^{2 } + 1}.}$$
     
  4. (4)

    Every point of the interval [0,π] is a limit point of the sequence Im  δ n , and every point of the intervals \([\log \,\rho _{1},\log \,\rho _{2}]\) , \([\log \,1/\rho _{2},\log \,1/\rho _{1}]\) is a limit point of the sequence Re  δ n .

     
  5. (5)
    There exist R = R(γ) > 0 and \(N = N(\gamma ) \in \mathbb{N}\) such that there is no point of the sequence \(\zeta _{k} = \sqrt{z_{k}}\) in the disks
    $$\displaystyle{ \left \vert \zeta -\pi \mathit{in}\right \vert < R,\;n > N. }$$
    (4.6.26)
     

It remains to consider the distribution of zeros of the function E α, β in the case α > 2.

Theorem 4.11 ([PopSed11, p. 45]).

Let α > 2. Then all zeros z n of the Mittag-Leffler function E α,β with sufficiently large modulus are simple and the following asymptotic formula holds:
$$\displaystyle{ z_{n} = \left ( \frac{\pi } {\sin \,\pi /\alpha }\left (n -\frac{1} {2} -\frac{\beta -1} {\alpha } \right ) +\alpha _{n}\right )^{\alpha }, }$$
(4.6.27)
where the sequence α n is defined as described below:
  1. (1)
    If the pair (α,β) is not mentioned in(4.6.4)and(4.6.5),then
    $$\displaystyle{\alpha _{n} = O\left (\mathrm{e}^{-\pi n(\cos \frac{\pi }{\alpha }-\cos \frac{3\pi } {\alpha } )/\sin \frac{\pi }{\alpha } }\right ).}$$
     
  2. (2)
    If 2 < α < 4, then
    $$\displaystyle{\alpha _{n} = O\left (n^{-\alpha \mathrm{Re}\,\tau _{\beta }}\mathrm{e}^{-\pi n\cot \frac{\pi }{\alpha } }\right ).}$$
     
  3. (3)
    If α ≥ 4, then
    $$\displaystyle{\alpha _{n} =\mathrm{ e}^{-\pi n\cot \frac{\pi }{\alpha } }\left (O\left (\mathrm{e}^{\pi n\cos \frac{3\pi } {\alpha } /\sin \frac{\pi }{\alpha } }\right ) + O\left (n^{-\alpha \mathrm{Re}\,\tau _{\beta }}\right )\right ).}$$
     

If β is real then all zeros z n with sufficiently large modulus are real too.

In a series of articles by different authors the following question, which is very important for applications, was discussed: “Are all zeros of the Mittag-Leffler function E α, β with α > 2 simple and negative?” This question goes back to an article by Wiman [Wim05b]. Several attempts to answer this question have shown its non-triviality. In [OstPer97] this question was reformulated as the following problem: “For any α ≥ 2, find a set W α consisting of those values of the positive parameter β such that all zeros of the Mittag-Leffler function E α, β are simple and negative.”

Let us give some answers to the above question, following [PopSed11].

Theorem 4.12.

For any α > 2, β ∈ (0,2α − 1], all complex zeros \((z_{n}(\alpha,\beta ))_{n\in \mathbb{N}}\) of the Mittag-Leffler function E α,β are simple and negative and satisfy the following inequalities
$$\displaystyle\begin{array}{rcl} -\xi _{1}^{\alpha }(\alpha,\beta ) < z_{ 1}(\alpha,\beta ) < -\frac{\varGamma (\alpha +\beta )} {\varGamma (\beta )},& &{}\end{array}$$
(4.6.28)
$$\displaystyle\begin{array}{rcl} -\xi _{n}^{\alpha }(\alpha,\beta ) < z_{ n}(\alpha,\beta ) < -\xi _{n-1}^{\alpha }(\alpha,\beta ),\;n \geq 2,& &{}\end{array}$$
(4.6.29)
where
$$\displaystyle{\xi _{n}^{\alpha }(\alpha,\beta ) = \frac{\pi \left (n + \frac{\beta -1} {\alpha } \right )} {\sin \frac{\pi }{\alpha }}.}$$

If α ≥ 4, then all zeros are simple and negative for any β ∈ (0,2α].

Theorem 4.13.

Let α ≥ 6, 0 < β ≤ 2α. Then for all \(n,1 \leq n \leq \left [ \frac{\alpha }{3}\right ] - 1\) , the zeros z n (α,β) of the Mittag-Leffler function satisfy the following inequalities
$$\displaystyle{ -\sqrt{2} \frac{\varGamma (\alpha n+\beta )} {\varGamma (\alpha (n - 1)+\beta )} < z_{n}(\alpha,\beta ) < - \frac{\varGamma (\alpha n+\beta )} {\varGamma (\alpha (n - 1)+\beta )}. }$$
(4.6.30)

Theorem 4.14.

For any \(N \in \mathbb{N}\) , N ≥ 3, the zeros z n (N,N + 1) of the Mittag-Leffler function E N,N+1 (z) satisfy the relation
$$\displaystyle{ z_{n}(N,N + 1) = -\left [\frac{\pi n +\pi /2 +\alpha _{n}(N)} {\sin \,\pi /\alpha } \right ]^{\alpha },\;n \in \mathbb{N},\;n \geq \left [N/3\right ], }$$
(4.6.31)
where \(\alpha _{n}(N) \in \mathbb{R},\vert \alpha _{n}(N)\vert \leq x_{n}(N)\) ,
$$\displaystyle{x_{n}(N) = \left \{\begin{array}{ll} \mathrm{exp}\{ -\pi n\cot \pi /\alpha \}, &3 \leq N \leq 6,\\ & \\ \mathrm{exp}\{ - 2\pi n\sin 2\pi /\alpha \}, &7 \leq N \leq 1,400,\\ & \\ 1.01\mathrm{exp}\{ - 2\pi n\sin 2\pi /\alpha \},&N > 1,400. \end{array} \right.}$$
If N ≥ 6, 1 ≤ n ≤ [N∕3] − 1, then
$$\displaystyle\begin{array}{rcl} & -\frac{((n+1)N)!} {(nN)!} \left (1 + \frac{3/2[((n+1)N)!]^{2}} {(nN)!((n+2)N)!}\min \{1,Nn^{-2}\}\right )& \\ & < z_{n}(N,N + 1) < -\frac{((n+1)N)!} {(nN)!}. &{}\end{array}$$
(4.6.32)

Next we present a few non-asymptotic results on the distribution of zeros of the Mittag-Leffler function.

Theorem 4.15.

Let 0 < α < 1. Then
  1. (1)

    For \(\beta \in \left (\bigcup \limits _{n=0}^{\infty }[-n+\alpha,-n + 1]\right )\bigcup [1,+\infty )\) the function E α,β (z) has no negative zero;

     
  2. (2)

    For \(\beta \in \bigcup \limits _{n=0}^{\infty }(-n,-n+\alpha )\) the function E α,β (z) has one negative zero and it is simple.

     

Theorem 4.16.

  1. (I)
    Let 0 < α < 1, β < 0. Then
    1. (1)

      For \(\beta \in [-2n - 1,-2n),n \in \mathbb{Z}_{+}\) , the function E α,β (z) has one positive zero and it is simple;

       
    2. (2)

      For \(\beta \in [-2n,-2n + 1),n \in \mathbb{N}\) , the set of zeros of the function E α,β (z) is either empty, or consists of two simple points, or consists of one double point.

       
     
  2. (II)

    The function E 1,β (z) has a unique simple positive zero, whenever \(\beta \in (-2n - 1,-2n),\;n \in \mathbb{Z}_{+}\) , and has no positive zero, whenever \(\beta \in (-2n,-2n + 1),\;n \in \mathbb{N}\) .

     

Theorem 4.17.

  1. (I)
    Let one of the following conditions be satisfied:
    1. (1)

      0 < α < 1,   β ∈ [1,1 + α],

       
    2. (2)

      1 < α < 2,   β ∈ [α − 1,1] ∪ [α,2].

       

    Then all zeros of the function E α,β (z) are located outside of the angle \(\vert \mathrm{arg}\,z\vert \leq \frac{\pi \alpha } {2}\) .

     
  2. (II)

    Let 1 < α < 2,β = 0. Then all zeros (≠0) are located outside of the angle \(\vert \mathrm{arg}\,z\vert \leq \frac{\pi \alpha } {2}\) .

     

4.7 Computations with the Two-Parametric Mittag-Leffler Function

Mittag-Leffler type functions play a basic role in the solution of fractional differential equations and integral equations of Abel type. Therefore, it seems important as a first step to develop their theory and stable methods for their numerical computation.

Integral representations play a prominent role in the analysis of entire functions. For the two-parametric Mittag-Leffler function (4.1.1) such representations in the form of an improper integral along the Hankel loop have been treated in the case β = 1 and in the general case with arbitrary β by Erdélyi et al. [Bat-3] and Dzherbashyan [Dzh54a, Dzh66]. They considered the representations
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) = \frac{1} {2\pi i\alpha }\int _{\gamma (\epsilon;\delta )}\frac{\mathrm{e}^{\zeta ^{1/\alpha } }\zeta ^{(1-\beta )/\alpha }} {\zeta -z} \mathrm{d}\zeta,\;\;\;z \in G^{(-)}(\epsilon;\delta ),& &{}\end{array}$$
(4.7.1)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) = \frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } } + \frac{1} {2\pi i\alpha }\int _{\gamma (\epsilon;\delta )}\frac{\mathrm{e}^{\zeta ^{1/\alpha } }\zeta ^{(1-\beta )/\alpha }} {\zeta -z} \mathrm{d}\zeta,\;\;\;z \in G^{(+)}(\epsilon;\delta ),& &{}\end{array}$$
(4.7.2)
under the conditions
$$\displaystyle{ 0 <\alpha < 2,\pi \alpha /2 <\delta <\min \{\pi,\pi \alpha \}. }$$
(4.7.3)
The contour γ(ε; δ) consists of two rays S δ (arg ζ = −δ, | ζ | ≥ ε) and S δ (arg ζ = δ, | ζ | ≥ ε) and a circular arc C δ (0; ε) ( | ζ |  = ε, −δ ≤ arg ≤ δ). On its left side there is a region G (−)(ε, δ), on its right side a region G (+)(ε, δ).
Using the integral representations in (4.7.1) and (4.7.2) it is not difficult to obtain asymptotic expansions for the Mittag-Leffler function in the complex plane (see Theorems 4.3 and 4.4). Let 0 < α < 2, β be an arbitrary number, and δ be chosen to satisfy the condition (4.7.3). Then we have, for any \(p \in \mathbb{N}\) (and for p = 0 if the “empty sum convention” is adopted) and | z | → 
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } } -\sum \limits _{k=1}^{p} \frac{z^{-k}} {\varGamma (\beta -\alpha k)} + O\left (\vert z\vert ^{-1-p}\right ),\,\forall z,\vert \mathrm{arg}\,z\vert \leq \delta. }$$
(4.7.4)
Analogously, for all z, δ ≤ | arg z | ≤ π, we have
$$\displaystyle{ E_{\alpha,\beta }(z) = -\sum \limits _{k=1}^{p} \frac{z^{-k}} {\varGamma (\beta -\alpha k)} + O\left (\vert z\vert ^{-1-p}\right ). }$$
(4.7.5)
These formulas are used in the numerical algorithm presented in this section (proposed in [GoLoLu02]). In what follows attention is restricted to the case \(\beta \in \mathbb{R}\), the most important one in the applications. For the purpose of numerical computation we look for integral representations better suited than (4.7.1) and (4.7.2). Denoting
$$\displaystyle{\phi (\zeta,z) = \frac{\mathrm{e}^{\zeta ^{1/\alpha } }\zeta ^{(1-\beta )/\alpha }} {\zeta -z} }$$
we represent the integral in formulas (4.7.1) and (4.7.2) in the form
$$\displaystyle\begin{array}{rcl} & I = \frac{1} {2\pi i\alpha }\int _{\gamma (\epsilon;\delta )}\phi (\zeta,z)\mathrm{d}\zeta = \frac{1} {2\pi i\alpha }\int _{S_{-\delta }}\phi (\zeta,z)\mathrm{d}\zeta & \\ & + \frac{1} {2\pi i\alpha }\int _{C_{\delta }(0;\epsilon )}\phi (\zeta,z)\mathrm{d}\zeta + \frac{1} {2\pi i\alpha }\int _{S_{\delta }}\phi (\zeta,z)\mathrm{d}\zeta = I_{1} + I_{2} + I_{3}.&{}\end{array}$$
(4.7.6)
The integrals I 1, I 2 and I 3 have to be transformed. For I 1 we take \(\zeta = r\mathrm{e}^{-i\delta }\), ε ≤ r < , and get
$$\displaystyle{ I_{1} = \frac{1} {2\pi i\alpha }\int _{S_{-\delta }}\phi (\zeta,z)\mathrm{d}\zeta = \frac{1} {2\pi i\alpha }\int _{+\infty }^{\epsilon }\frac{\mathrm{e}^{(r\mathrm{e}^{-i\delta }) ^{1/\alpha } }(r\mathrm{e}^{-i\delta })^{(1-\beta )/\alpha }} {(r\mathrm{e}^{-i\delta }) - z} \mathrm{e}^{-i\delta }\mathrm{d}r. }$$
(4.7.7)
Analogously, by using \(\zeta = r\mathrm{e}^{i\delta }\), ε ≤ r < ,
$$\displaystyle{ I_{3} = \frac{1} {2\pi i\alpha }\int _{S_{\delta }}\phi (\zeta,z)\mathrm{d}\zeta = \frac{1} {2\pi i\alpha }\int _{\epsilon }^{+\infty }\frac{\mathrm{e}^{(r\mathrm{e}^{i\delta }) ^{1/\alpha } }(r\mathrm{e}^{i\delta })^{(1-\beta )/\alpha }} {(r\mathrm{e}^{i\delta }) - z} \mathrm{e}^{i\delta }\mathrm{d}r. }$$
(4.7.8)
For I 2 with \(\zeta =\epsilon \mathrm{ e}^{i\varphi },\;-\delta \leq \varphi \leq \delta\)
$$\displaystyle\begin{array}{rcl} & I_{2} = \frac{1} {2\pi i\alpha }\int _{C_{\delta }(0;\epsilon )}\phi (\zeta,z)\mathrm{d}\zeta = \frac{1} {2\pi i\alpha }\int _{-\delta }^{\delta }\frac{\mathrm{e}^{(\epsilon \mathrm{e}^{i\varphi })^{1/\alpha }}(\epsilon \mathrm{e}^{i\varphi })^{(1-\beta )/\alpha }} {(\epsilon \mathrm{e}^{i\varphi })-z} \epsilon i\mathrm{e}^{i\varphi }\mathrm{d}\varphi & \\ & = \frac{\epsilon ^{1+(1-\beta )/\alpha }} {2\pi \alpha } \int _{-\delta }^{\delta }\frac{\mathrm{e}^{\epsilon ^{1/\alpha }(\mathrm{e}^{i\varphi /\alpha }) }\mathrm{e}^{(i\varphi (1-\beta )/\alpha +1)}} {\epsilon \mathrm{e}^{i\varphi }-z} \mathrm{d}\varphi =\int _{ -\delta }^{\delta }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi,&{}\end{array}$$
(4.7.9)
where
$$\displaystyle\begin{array}{rcl} & P[\alpha,\beta,\epsilon,\varphi,z] = \frac{\epsilon ^{1+(1-\beta )/\alpha }} {2\pi \alpha } \frac{\mathrm{e}^{\epsilon ^{1/\alpha }\cos \,(\varphi /\alpha ) }(\cos \,(\omega )+i\sin \,(\omega ))} {\epsilon \mathrm{e}^{i\varphi }-z},& \\ & \omega =\epsilon ^{1/\alpha }\sin \,(\varphi /\alpha ) +\varphi (1 + (1-\beta )/\alpha ). &{}\end{array}$$
(4.7.10)
The sum I 1 and I 3 can be rewritten as
$$\displaystyle{ I_{1} + I_{3} =\int _{ \epsilon }^{+\infty }K[\alpha,\beta,\delta,r,z]\mathrm{d}r, }$$
(4.7.11)
where
$$\displaystyle\begin{array}{rcl} & K[\alpha,\beta,\delta,r,z] = \frac{1} {2\pi \alpha }r^{(1-\beta )/\alpha }\mathrm{e}^{r^{1/\alpha }\cos \,(\delta /\alpha ) } \frac{r\sin \,(\psi -\delta )-z\sin \,(\psi )} {r^{2}-2\mathit{rz}\cos \,(\delta )+z^{2}},& \\ & \psi = r^{1/\alpha }\sin \,(\delta /\alpha ) +\varphi (1 + (1-\beta )/\alpha ). &{}\end{array}$$
(4.7.12)
Using the above notation formulas (4.7.1) and (4.7.2) can be rewritten in the form
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\int _{ \epsilon }^{+\infty }K[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}r +\int _{ -\delta }^{\delta }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi,\;\;\;z \in G^{(-)}(\epsilon;\delta ),& &{}\end{array}$$
(4.7.13)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\int _{ \epsilon }^{+\infty }K[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}r +\int _{ -\delta }^{\delta }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi & & \\ +\frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } },\;\;\;z \in G^{(+)}(\epsilon;\delta ).& &{}\end{array}$$
(4.7.14)
Let us now consider the case 0 < α ≤ 1, z ≠ 0. By condition (4.7.3) we can choose δ = min {π, π α} = π α. Then the kernel function (4.7.12) looks simpler:
$$\displaystyle\begin{array}{rcl} K[\alpha,\beta,\pi \alpha,r,z] = \tilde{K}[\alpha,\beta,r,z]& & \\ = \frac{1} {2\pi \alpha }r^{(1-\beta )/\alpha }\mathrm{e}^{-r^{1/\alpha } }\frac{r\sin \,(\pi (1-\beta )) - z\sin \,(\pi (1 -\beta +\alpha )} {r^{2} - 2\mathit{rz}\cos \,(\pi \alpha ) + z^{2}}.& &{}\end{array}$$
(4.7.15)
We distinguish three possibilities for arg z in the formulas (4.7.13)–(4.7.15) for the computation of the function E α, β (z) at an arbitrary point \(z \in \mathbb{C},z\not =0\), namely
  1. (A)

     | arg z |  > π α;

     
  2. (B)

     | arg z |  = π α;

     
  3. (C)

     | arg z |  < π α.

     

The following theorems give representation formulas suitable for further numerical calculations.

Theorem 4.18.

Under the conditions
$$\displaystyle{0 <\alpha \leq 1,\;\beta \in \mathbb{R},\;\vert \mathrm{arg}\,z\vert >\pi \alpha,\;z\not =0,}$$
the function E α,β (z) has the representations
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\int _{ \epsilon }^{+\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r +\int _{ -\pi \alpha }^{\pi \alpha }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi,\;\epsilon > 0,\;\beta \in \mathbb{R},& &{}\end{array}$$
(4.7.16)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\int _{ 0}^{+\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r,\;\mathrm{if}\;\beta < 1+\alpha,& &{}\end{array}$$
(4.7.17)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) = -\frac{\sin \,(\pi \alpha )} {\pi \alpha } \int _{0}^{+\infty } \frac{\mathrm{e}^{-r^{1/\alpha } }} {r^{2} - 2\mathit{rz}\cos \,(\pi \alpha ) + z^{2}}\,\mathrm{d}r -\frac{1} {z},\;\mathrm{if}\;\beta = 1 +\alpha.& &{}\end{array}$$
(4.7.18)

Theorem 4.19.

Under the conditions
$$\displaystyle{0 <\alpha \leq 1,\;\beta \in \mathbb{R},\;\vert \mathrm{arg}\,z\vert =\pi \alpha,\;z\not =0,}$$
the function E α,β (z) has the representations
$$\displaystyle{ E_{\alpha,\beta }(z) =\int _{ \epsilon }^{+\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r +\int _{ -\pi \alpha }^{\pi \alpha }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi,\;\epsilon > \vert z\vert, }$$
(4.7.19)
where the kernel functions \(\tilde{K}[\alpha,\beta,r,z]\) and \(P[\alpha,\beta,\epsilon,\varphi,z]\) are given by the formulas(4.7.15) and (4.7.10),respectively.

Theorem 4.20.

Under the conditions
$$\displaystyle{0 <\alpha \leq 1,\;\beta \in \mathbb{R},\;\vert \mathrm{arg}\,z\vert <\pi \alpha,\;z\not =0,}$$
the function E α,β (z) has the representations
$$\displaystyle\begin{array}{rcl} & E_{\alpha,\beta }(z) =\int _{ \epsilon }^{+\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r +\int _{ -\pi \alpha }^{\pi \alpha }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi & \\ & +\frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } },\;0 <\epsilon < \vert z\vert,\;\beta \in \mathbb{R}; &{}\end{array}$$
(4.7.20)
$$\displaystyle\begin{array}{rcl} E_{\alpha,\beta }(z) =\int _{ 0}^{+\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r + \frac{1} {\alpha } z^{(1-\beta )/\alpha }\mathrm{e}^{z^{1/\alpha } },\;\mathrm{if}\;\beta < 1+\alpha;& &{}\end{array}$$
(4.7.21)
$$\displaystyle\begin{array}{rcl} & E_{\alpha,\beta }(z) = -\frac{\sin \,(\pi \alpha )} {\pi \alpha } \int _{0}^{+\infty } \frac{\mathrm{e}^{-r^{1/\alpha }}} {r^{2}-2\mathit{rz}\cos \,(\pi \alpha )+z^{2}} \,\mathrm{d}r& \\ & -\frac{1} {z} + \frac{1} {\alpha z}\mathrm{e}^{z^{1/\alpha } },\;\mathrm{if}\;\beta = 1+\alpha, &{}\end{array}$$
(4.7.22)
where the kernel functions \(\tilde{K}[\alpha,\beta,r,z]\) and \(P[\alpha,\beta,\epsilon,\varphi,z]\) are given by formulas(4.7.15)and(4.7.10),respectively.
Therefore, for arbitrary z ≠ 0 and 0 < α ≤ 1 the Mittag-Leffler function E α, β (z) can be represented by one of the formulas (4.7.16)–(4.7.22). These formulas are used for numerical computation if q <  | z | , 0 < q < 1 and 0 < α ≤ 1. In the case | z | ≤ q, 0 < q < 1, the values of the Mittag-Leffler function are computed for arbitrary α > 0 by using series representation (4.1.1). The case α > 1 is reduced to the case 0 < α ≤ 1 by using recursion formulas. To compute the function E α, β (z) for arbitrary \(z \in \mathbb{C}\) with arbitrary indices \(\alpha > 0,\beta \in \mathbb{R}\), three possibilities are distinguished:
  1. (A)

     | z | ≤ q, 0 < q < 1 (q is a fixed number), α > 0;

     
  2. (B)

     | z |  > q, 0 < α ≤ 1;

     
  3. (C)

     | z |  > q, α > 1.

     

In each case the Mittag-Leffler function can be computed with the prescribed accuracy ρ > 0. In the case (A) the computations are based on the following result:

Theorem 4.21.

In the case (A) the Mittag-Leffler function can be computed with the prescribed accuracy ρ > 0 by use of the formula
$$\displaystyle{ E_{\alpha,\beta }(z) =\sum \limits _{ k=0}^{k_{0} } \frac{z^{k}} {\varGamma (\alpha k+\beta )} +\mu (z),\;\;\;\vert \mu (z)\vert <\rho, }$$
(4.7.23)
where
$$\displaystyle{k_{0} =\max \,\{ [(1-\beta )/\alpha ] + 1;[\ln \,(\rho (1 -\vert z\vert ))/\ln \,(\vert z\vert )]\}.}$$
In the case (B) one can use the integral representations (4.7.16)–(4.7.22). For this it is necessary to compute numerically either the improper integral
$$\displaystyle{I =\int \limits _{ a}^{\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r,\;\;\;a \in \{ 0;\epsilon \},}$$
and/or the integral
$$\displaystyle{J =\int \limits _{ -\pi \alpha }^{\pi \alpha }P[\alpha,\beta,\epsilon,\varphi,z]\mathrm{d}\varphi,\;\;\;\epsilon > 0.}$$
To calculate the first (improper) integral I of the bounded function \(\tilde{K}[\alpha,\beta,r,z]\) the following theorem is used:

Theorem 4.22.

The representation
$$\displaystyle{ I =\int _{ a}^{\infty }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r =\int _{ a}^{r_{0} }\tilde{K}[\alpha,\beta,r,z]\mathrm{d}r +\mu (r),\;\;\;\vert \mu (r)\vert \leq \rho,\;\;\;a \in \{ 0;\epsilon \}, }$$
(4.7.24)
is valid under the conditions
$$\displaystyle\begin{array}{rcl} & 0 <\alpha \leq 1,\;\;\;\vert z\vert > q > 0, & {}\\ & r_{0} = \left \{\begin{array}{lll} \max \,\{1,2\vert z\vert,(-\ln \,(\pi \rho /6))^{\alpha }\}, &\mathrm{if}&\beta \geq 0;\\ \\ \max \,\{(1 + \vert \beta \vert )^{\alpha },2\vert z\vert,(-2\ln \,(\pi \rho /(6(\vert \beta \vert + 2)(2\vert \beta \vert )^{\vert \beta \vert })))^{\alpha }\},&\mathrm{if}&\beta < 0. \end{array} \right.& {}\\ \end{array}$$

The second integral J (the integrand \(P[\alpha,\beta,\epsilon,\varphi,z]\) being bounded and the limits of integration being finite) can be calculated with prescribed accuracy ρ > 0 by one of many product quadrature methods.

In the case (C) the following recursion formula is used (see [Dzh66])
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {m}\sum \limits _{l=0}^{m-1}E_{\alpha /m,\beta }(z^{1/m}\mathrm{e}^{2\pi \mathit{il}/m}),\;\;\;m \geq 1. }$$
(4.7.25)
In order to reduce case (C) to the cases (B) and (A) one can take m = [α] + 1 in formula (4.7.25). Then 0 < αm. 1, and we calculate the functions \(E_{\alpha /m,\beta }(z^{1/m}\mathrm{e}^{2\pi \mathit{il}/m})\) as in case (A) if | z | 1∕m  ≤ q < 1, and as in case (B) if | z | 1∕m  > q.

Remark 4.23.

The ideas and techniques employed for the Mittag-Leffler function can be used for numerical calculation of other functions of hypergeometric type. In particular, the same method with some small modifications can be applied to the Wright function, which plays a very important role in the theory of partial differential equations of fractional order (see, e.g., [BucLuc98, GoLuMa00, Luc00, LucGor98, MaLuPa01]). To this end, the following representations of the Wright function (see [GoLuMa99]) can be used in place of the corresponding representations of the Mittag-Leffler function:
$$\displaystyle\begin{array}{rcl} & \phi (\rho,\beta;z) =\sum \limits _{ k=0}^{\infty } \frac{z^{k}} {\varGamma (\rho k+\beta )},\;\;\;\rho > -1,\;\;\;\beta \in \mathbb{C}, & {}\\ & \phi (\rho,\beta;z) = \frac{1} {2\pi i}\int _{\mathit{Ha}}\mathrm{e}^{\zeta +z\zeta ^{-\rho } }\zeta ^{-\beta }\mathrm{d}\zeta,\;\;\;\rho > -1,\;\;\;\beta \in \mathbb{C},& {}\\ \end{array}$$
where Ha denotes the Hankel path in the ζ-plane with a cut along the negative real semi-axis arg ζ = π.

4.8 Extension for Negative Values of the First Parameter

The two-parametric Mittag-Leffler function (4.1.1), defined in the form of a series, exists only for the values of parameters Re α > 0 and \(\beta \in \mathbb{C}\). However, by using an existing integral representation formula for the two-parametric Mittag-Leffler function (see, e.g., [Dzh66, KiSrTr06]) it is possible to determine an extension of the two-parametric Mittag-Leffler function to other values of the first parameter.

In this section we present an analytic continuation of the Mittag-Leffler function depending on real parameters \(\alpha,\beta \in \mathbb{R}\) by extending its domain to negative \(\alpha < 0\). Here we follow the results of [Han-et-al09].

The following integral representation of the Mittag-Leffler function is known (see, e.g., [Dzh66, KiSrTr06])
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {2\pi }\int _{\mathit{Ha}} \frac{t^{\alpha -\beta }\mathrm{e}^{t}} {t^{\alpha } - z}\mathrm{d}t,\;\;\;z \in \mathbb{C}, }$$
(4.8.1)
where the contour of integration Ha is the so-called Hankel path, a loop starting and ending at −, and encircling the disk | t | ≤ | z | 1∕α counterclockwise.
To find an equation which can determine E α, β (z), we rewrite the integral representation of the Mittag-Leffler function (4.8.1) as
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {2\pi }\int _{\mathit{Ha}} \frac{\mathrm{e}^{t}} {t^{\beta } - zt^{-\alpha +\beta }}\mathrm{d}t }$$
(4.8.2)
and expand part of the integrand in (4.8.2) in partial fractions as follows:
$$\displaystyle{ \frac{1} {t^{\beta } - zt^{-\alpha +\beta }} = \frac{1} {t^{\beta }} - \frac{1} {t^{\beta } - z^{-1}t^{\alpha +\beta }}. }$$
(4.8.3)
Substituting Eq. (4.8.3) into (4.8.2) yields
$$\displaystyle{ E_{\alpha,\beta }(z) = \frac{1} {2\pi }\int _{\mathit{Ha}}\frac{\mathrm{e}^{t}} {t^{\beta }} \mathrm{d}t -\frac{1} {2\pi }\int _{\mathit{Ha}} \frac{\mathrm{e}^{t}} {t^{\beta } - z^{-1}t^{\alpha +\beta }}\mathrm{d}t,\;\;\;z \in \mathbb{C}\setminus \{0\}. }$$
(4.8.4)
This gives the following definition of the Mittag-Leffler function with negative value of the first parameter:
$$\displaystyle{ E_{-\alpha,\beta }(z) = \frac{1} {\varGamma (\beta )} - E_{\alpha,\beta }\left (\frac{1} {z}\right ),\;\;\;\alpha > 0,\;\beta \in \mathbb{R};z \in \mathbb{C}\setminus \{0\}. }$$
(4.8.5)
In particular,
$$\displaystyle{E_{-\alpha }(z):= E_{-\alpha,1}(z) = 1 - E_{\alpha }\left (\frac{1} {z}\right ),\;\;\;\alpha > 0;z \in \mathbb{C}\setminus \{0\}.}$$
By using the known recurrence formula
$$\displaystyle{E_{\alpha,\beta }(z) = \frac{1} {\varGamma (\beta )} + zE_{\alpha,\alpha +\beta }\left (z\right )}$$
we obtain another variant of the definition (4.8.5)
$$\displaystyle{ E_{-\alpha,\beta }(z) = -\frac{1} {z}E_{\alpha,\alpha +\beta }\left (\frac{1} {z}\right ),\;\;\;\alpha > 0,\;\beta \in \mathbb{R};z \in \mathbb{C}\setminus \{0\}. }$$
(4.8.6)

Direct calculations show that definitions (4.8.5) and (4.8.6) determine the same function, analytic in \(\mathbb{C}\setminus \{0\}\).

By taking the limit in (4.8.5) as α → +0 we get the definition of E 0, β (z)
$$\displaystyle{ E_{0,\beta }(z) = \frac{1} {\varGamma (\beta )(1 - z)},\;\;\;\beta \in \mathbb{R};\vert z\vert < 1. }$$
(4.8.7)
Obviously, this function can be analytically continued in the domain \(\mathbb{C}\setminus \{1\}\).
From the definition of the two-parametric Mittag-Leffler function (4.1.1) we obtain the following series representation of the extended Mittag-Leffler function (i.e. the function corresponding to negative values of the first parameter):
$$\displaystyle{ E_{-\alpha,\beta }(z) = -\sum \limits _{k=1}^{\infty } \frac{1} {\varGamma (\alpha z+\beta )}\left (\frac{1} {z}\right )^{k},\;\;\;z \in \mathbb{C}\setminus \{0\}. }$$
(4.8.8)

By using this representation and the above definitions of the extended Mittag-Leffler function (4.8.5) (or (4.8.6)) one can obtain functional, differential and recurrence relations which are analogous to corresponding relations for the two-parametric function with positive first parameter.

Proposition 4.24.

Let α > 0, \(\beta \in \mathbb{R}\) . Then the following formulas are valid for all values of parameters for which all items are defined.
  1. A.
    Recurrence relations.
    $$\displaystyle\begin{array}{rcl} E_{-\alpha,\beta }(z) + E_{-\alpha,\beta }(-z) = 2E_{-2\alpha,\beta }(z^{2});& & {}\end{array}$$
    (4.8.9)
    $$\displaystyle\begin{array}{rcl} E_{-n\alpha,\beta }(z) = \frac{1} {n}\sum \limits _{k=0}^{n-1}E_{ -\alpha,\beta }(z\mathrm{e}^{-2\pi \mathit{ik}/n});& & {}\end{array}$$
    (4.8.10)
    $$\displaystyle\begin{array}{rcl} E_{-\alpha,\beta }(z) = z^{n}E_{ -\alpha,\beta -\alpha n}(z) +\sum \limits _{ k=0}^{n-1} \frac{z^{k}} {\varGamma (\beta -\alpha k)};& & {}\end{array}$$
    (4.8.11)
    $$\displaystyle\begin{array}{rcl} E_{-\alpha }(-z) = E_{-2\alpha }(z^{2}) + E_{ -2\alpha }(z^{2}) - zE_{ -2\alpha,\alpha +1}(z^{2}).& & {}\end{array}$$
    (4.8.12)
     
  2. B.
    Differential relations.
    $$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}} {\mathrm{d}z}\left [z^{1-\beta }E_{ -\alpha,\beta }(z^{\alpha })\right ] = -z^{-\beta }E_{ -\alpha,\beta -1}(z^{\alpha });& & {}\end{array}$$
    (4.8.13)
    $$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}} {\mathrm{d}z}\left [E_{-\alpha }(z)\right ] = - \frac{1} {\varGamma (\alpha +1)} + \frac{1} {\alpha } E_{-\alpha,\alpha }(z);& & {}\end{array}$$
    (4.8.14)
    $$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}^{n}} {\mathrm{d}z^{n}}\left [E_{-n}(z^{-n})\right ] = E_{ -n}(z^{-n}).& & {}\end{array}$$
    (4.8.15)
     
  3. C.
    Functional relations.
    $$\displaystyle\begin{array}{rcl} \int \limits _{0}^{z}E_{ -\alpha,\beta }(t^{\alpha })t^{-\beta -1}\mathrm{d}t = -z^{-\beta }E_{ -\alpha,\beta +1}(z^{\alpha });& & {}\end{array}$$
    (4.8.16)
    $$\displaystyle\begin{array}{rcl} \mathcal{L}\left [z^{\beta -1}E_{ -\alpha,\beta }\left ( \frac{1} {\pm \mathit{az}^{\alpha }}\right )\right ] = \frac{\mp a} {s^{\beta }(s^{\alpha } \mp a)}.& & {}\end{array}$$
    (4.8.17)
     

4.9 Further Analytic Properties

Here we present a number of integral and differential formulas for the Mittag-Leffler function. The integral relations below can easily be established by applying classical formulas for Gamma and Beta functions (see Appendix A) and other techniques.
$$\displaystyle\begin{array}{rcl} \int \limits _{0}^{\infty }\mathrm{e}^{-x}x^{\beta -1}E_{\alpha,\beta }\left (x^{\alpha }z\right )\mathrm{d}x = \frac{1} {1 - z}\;\;\;(\vert z\vert < 1,\;\alpha,\beta \in \mathbb{C},\;\mathrm{Re}\,\alpha > 0,\;\mathrm{Re}\,\beta > 0).& &{}\end{array}$$
(4.9.1)
$$\displaystyle\begin{array}{rcl} \int \limits _{0}^{x}\left (x-\zeta \right )^{\beta -1}E_{\alpha }\left (\zeta ^{\alpha }\right )\mathrm{d}\zeta =\varGamma (\beta )x^{\beta }E_{\alpha,\beta +1}(x^{\alpha })\;\;\;(\alpha,\beta \in \mathbb{C},\;\mathrm{Re}\,\alpha >0,\;\mathrm{Re}\,\beta >0).& &{}\end{array}$$
(4.9.2)
$$\displaystyle\begin{array}{rcl} \int \limits _{0}^{\infty }\mathrm{e}^{-\mathit{sx}}x^{m\alpha +\beta -1}E_{\alpha,\beta }^{(m)}\left (\pm \lambda x^{\alpha }\right )\mathrm{d}x = \frac{m!S^{\alpha -\beta }} {s^{\alpha }\mp \lambda },& &{}\end{array}$$
(4.9.3)
where \(\vert \lambda s^{-\alpha }\vert < 1,\;\alpha,\beta \in \mathbb{C},\;\mathrm{Re}\,\alpha > 0,\;\mathrm{Re}\,\beta > 0\).
$$\displaystyle\begin{array}{rcl} & \int \limits _{0}^{x}\zeta ^{\beta _{1}-1}E_{\alpha,\beta _{ 1}}\left (\lambda \zeta ^{\alpha }\right )(x-\zeta )^{\beta _{2}-1}E_{\alpha,\beta _{2}}\left (\mu (x-\zeta )^{\alpha }\right )\mathrm{d}\zeta & \\ & = \frac{x^{\beta _{1}+\beta _{2}-1}} {\lambda -\mu } \left \{E_{\alpha,\beta _{1}+\beta _{2}}\left (\lambda x^{\alpha }\right ) - E_{\alpha,\beta _{1}+\beta _{2}}\left (\mu x^{\alpha }\right )\right \}.&{}\end{array}$$
(4.9.4)
The differential relations below follow by direct calculation:
$$\displaystyle\begin{array}{rcl} \left ( \frac{\partial } {\partial z}\right )^{n}\,\left [z^{\beta -1}E_{\alpha,\beta }(\lambda z^{\alpha })\right ] = z^{\beta -n-1}E_{\alpha,\beta -n}(\lambda z^{\alpha }).& &{}\end{array}$$
(4.9.5)
$$\displaystyle\begin{array}{rcl} \left (\frac{\partial } {\partial \lambda }\right )^{n}\,\left [z^{\beta -1}E_{\alpha,\beta }(\lambda z^{\alpha })\right ] = n!z^{\alpha n+\beta -1}E_{\alpha,\alpha n+\beta }^{n+1}(\lambda z^{\alpha }),& &{}\end{array}$$
(4.9.6)
where \(E_{\alpha,\alpha n+\beta }^{n+1}(z)\) is the Prabhakar three-parametric function (see Sect.  5.1 below).
A special case of the two-parametric Mittag-Leffler function (the so-called α-exponential function ) is of interest for many applications. It is defined in the following way:
$$\displaystyle{ e_{\alpha }^{\lambda z}:= z^{\alpha -1}E_{\alpha,\alpha }(\lambda z^{\alpha })\;\;\;(z \in \mathbb{C}\setminus \{0\},\lambda \in \mathbb{C}). }$$
(4.9.7)
For all \(\alpha \in \mathbb{C},\,\mathrm{Re}\,\alpha > 0\), it can be represented in the form of the series
$$\displaystyle{ e_{\alpha }^{\lambda z} = z^{\alpha -1}\sum \limits _{ k=0}^{\infty }\lambda ^{k} \frac{z^{\alpha k}} {\varGamma \left ((k + 1)\alpha \right )}, }$$
(4.9.8)
which converges in \(\mathbb{C}\setminus \{0\}\) and determines in this domain an analytic function. The simple properties of this function:
$$\displaystyle\begin{array}{rcl} (1)\;\lim _{z\rightarrow 0}z^{1-\alpha }e_{\alpha }^{\lambda z} = \frac{1} {\varGamma (\alpha )}\;\;\;(\mathrm{Re}\,\alpha > 0),& &{}\end{array}$$
(4.9.9)
$$\displaystyle\begin{array}{rcl} (2)\;e_{1}^{\lambda z} =\mathrm{ e}^{\lambda z},& &{}\end{array}$$
(4.9.10)
justify its name. However, the α-exponential function does not satisfy the main property of the exponential function, i.e.,
$$\displaystyle{ e_{\alpha }^{\lambda z}e_{\alpha }^{\mu z}\not =e_{\alpha }^{(\lambda +\mu )z}. }$$
(4.9.11)
For 0 < α < 2, the α-exponential function satisfies a simple asymptotic relation
$$\displaystyle{ e_{\alpha }^{\lambda z} = \frac{\lambda ^{(1-\alpha )/\alpha }} {\alpha } \mbox{ exp}\{\lambda ^{1/\alpha }z\} -\sum \limits _{ k=1}^{N-1}\frac{\lambda ^{-k-1}} {\varGamma (-\alpha k)} \frac{1} {z^{\alpha k+1}} + O\left ( \frac{1} {z^{\alpha N+1}}\right ), }$$
(4.9.12)
where z → , \(N \in \mathbb{N}\setminus \{1\}\), | arg(λ z α ) | ≤ μ, \(\frac{\pi \alpha }{2} <\mu <\min \{\pi,\pi \alpha \}\), and
$$\displaystyle{ e_{\alpha }^{\lambda z} = -\sum \limits _{ k=1}^{N-1}\frac{\lambda ^{-k-1}} {\varGamma (-\alpha k)} \frac{1} {z^{\alpha k+1}} + O\left ( \frac{1} {z^{\alpha N+1}}\right ), }$$
(4.9.13)
where z → , \(N \in \mathbb{N}\setminus \{1\}\), μ ≤ | arg(λ z α ) | ≤ π.

When α ≥ 2, the asymptotic behaviour at infinity of the α-exponential function is more complicated (see, e.g., [KiSrTr06, pp. 51–52]).

4.10 The Two-Parametric Mittag-Leffler Function of a Real Variable

4.10.1 Integral Transforms of the Two-Parametric Mittag-Leffler Function

The following form of the Laplace transform of the two-parametric Mittag-Leffler function is most often used in applications:
$$\displaystyle{ \left (\mathcal{L}\,t^{\beta -1}E_{\alpha,\beta }(\lambda t^{\alpha })\right )(s) = \frac{s^{\alpha -\beta }} {s^{\alpha }-\lambda }\;\;\;(\mathrm{Re}\,s > 0,\;\lambda \in \mathbb{C},\;\vert \lambda s^{-\alpha }\vert < 1). }$$
(4.10.1)
It can be shown directly that the Laplace transform of the two-parametric Mittag-Leffler function E α, β (t) is given in terms of the Wright function (see, e.g., [KiSrTr06, p. 44])
$$\displaystyle{ \left (\mathcal{L}\,E_{\alpha,\beta }(t)\right )(s) = \frac{1} {s}_{2}\varPsi _{1}\left [\begin{array}{l} (1,1),(1,1)\\ (\alpha,\beta ) \end{array} \bigg\vert \frac{1} {s}\right ]\;\;\;(\mathrm{Re}\,s > 0). }$$
(4.10.2)
From the Mellin–Barnes integral representation of the two-parametric Mittag-Leffler function we arrive at the following formula for the Mellin transform of this function
$$\displaystyle{ \left (\mathcal{M}\,E_{\alpha,\beta }(-t)\right )(s) =\int \limits _{ 0}^{\infty }E_{\alpha,\beta }(-t)t^{s-1}\mathrm{d}t = \frac{\varGamma (s)\varGamma (1 - s)} {\varGamma (\beta -\alpha s)} \;\;\;(0 < \mathrm{Re} < 1). }$$
(4.10.3)
To conclude this subsection, we consider the Fourier transform of the two-parametric Mittag-Leffler function E α, β ( | t | ) with α > 1. Performing a term-by-term integration of the series we get the formula (α > 1):
$$\displaystyle{ \left (\mathcal{F}\,E_{\alpha,\beta }(\vert t\vert )\right )(x):=\int \limits _{ -\infty }^{+\infty }\mathrm{e}^{\mathit{ixt}}E_{\alpha,\beta }(\vert t\vert )\mathrm{d}t = \frac{\delta (x)} {\varGamma (\beta )} -\frac{2} {x^{2}}_{2}\varPsi _{1}\left [\begin{array}{l} (2,2),(1,1)\\ (\alpha +\beta, 2\alpha ) \end{array} \bigg\vert - \frac{1} {x^{2}}\right ], }$$
(4.10.4)
where δ(⋅ ) is the Dirac delta function.
Since for all t
$$\displaystyle{ E_{\alpha,\beta }(\vert t\vert ) - \frac{1} {\varGamma (\beta )} = \vert t\vert E_{\alpha,\alpha +\beta }(\vert t\vert ), }$$
(4.10.5)
formula (4.10.4) can be simplified
$$\displaystyle{ \left (\mathcal{F}\,\vert t\vert E_{\alpha,\alpha +\beta }(\vert t\vert )\right )(x) = -\frac{2} {x^{2}}_{2}\varPsi _{1}\left [\begin{array}{l} (2,2),(1,1)\\ (\alpha +\beta, 2\alpha ) \end{array} \bigg\vert - \frac{1} {x^{2}}\right ]\;(\alpha > 1,\beta \in \mathbb{C}). }$$
(4.10.6)

4.10.2 The Complete Monotonicity Property

Let us show that the generalized Mittag-Leffler function E α, β (−x) possesses the complete monotonicity property for 0 ≤ α ≤ 1, β ≥ α. In fact, this result follows from the complete monotonicity of the classical Mittag-Leffler function E α (−x) due to the following technical lemmas.

Lemma 4.25.

For all α ≥ 0
$$\displaystyle{E_{\alpha,\alpha }(-x) = -\alpha \frac{\mathrm{d}} {\mathrm{d}x}E_{\alpha }(-x).}$$

\(\vartriangleleft \) This follows from the standard properties of the integral depending on a parameter. \(\vartriangleright \)

Lemma 4.26.

Let β > α > 0. Then the following identity holds:
$$\displaystyle{ E_{\alpha,\beta }(-x) = \frac{1} {\alpha \varGamma (\beta -\alpha )}\int \limits _{0}^{1}\left (1 - t^{1/\alpha }\right )^{\beta -\alpha -1}E_{\alpha,\alpha }(-\mathit{tx})\mathrm{d}t. }$$
(4.10.7)
\(\vartriangleleft \) Let us take E α, α (−tx) in the form of a series and substitute it into the right-hand side of (4.10.7). By interchanging the order of integration and summation (which can be easily justified) we obtain that the right-hand side is equal to
$$\displaystyle{ \frac{1} {\alpha \varGamma (\beta -\alpha )}\sum \limits _{k=0}^{\infty }\frac{(-x)^{k}} {\varGamma (\alpha k+\alpha )} \int \limits _{0}^{1}t^{k}\left (1 - t^{1/\alpha }\right )^{\beta -\alpha -1}\mathrm{d}t.}$$
Calculating these integrals we arrive at the series representation for E α, β (−x). \(\vartriangleright \)
Observe that
$$\displaystyle\begin{array}{rcl} & E_{0,\beta }(-x) = \frac{1} {\alpha \varGamma (\beta )} \frac{1} {1+x},\;\;\;\beta > 0,& {}\\ & E_{0,\beta }(-x) = 0,\;\;\;\beta = 0. & {}\\ \end{array}$$
In both cases E 0, β (−x) is completely monotonic.

The complete monotonicity of E α, β (−x) then follows immediately from Pollard’s result [Poll48], see Sect.  3.7.2 of this book.

4.10.3 Relations to the Fractional Calculus

Here we present a few formulas related to the values of the fractional integrals and derivatives of the two-parametric Mittag-Leffler function (see, e.g., [HaMaSa11, pp. 15–16]). Let us start with the left-sided Riemann–Liouville integral. Suppose that \(\mathrm{Re}\,\alpha > 0,\;\mathrm{Re}\,\beta > 0,\;\mathrm{Re}\,\gamma > 0,\;a \in \mathbb{R}\). Then by using the series representation and the left-sided Riemann–Liouville integral of the power function we get
$$\displaystyle{ \left (I_{0+}^{\alpha }\,t^{\gamma -1}E_{\beta,\gamma }(\mathit{at}^{\beta })\right )(x) = x^{\alpha +\gamma -1}\left (E_{\beta,\alpha +\gamma }(\mathit{ax}^{\beta })\right ), }$$
(4.10.8)
and, in particular, if a ≠ 0, then (for β = α)
$$\displaystyle{ \left (I_{0+}^{\alpha }\,t^{\gamma -1}E_{\alpha,\gamma }(\mathit{at}^{\alpha })\right )(x) = \frac{x^{\gamma -1}} {a} \left (E_{\alpha,\gamma }(\mathit{ax}^{\alpha }) - \frac{1} {\varGamma (\gamma )}\right ). }$$
(4.10.9)
In the same manner one can obtain the formula
$$\displaystyle{ \left (I_{0+}^{\alpha }\,t^{\alpha -1}E_{\alpha,\beta }(\mathit{at}^{\alpha })\right )(x) = \frac{x^{\alpha -1}} {a} \left (E_{\alpha,\beta }(\mathit{ax}^{\alpha }) - \frac{1} {\varGamma (\beta )}\right ). }$$
(4.10.10)
Analogously, one can calculate the right-sided fractional Riemann–Liouville integral of the two-parametric Mittag-Leffler function in the case \(\mathrm{Re}\,\alpha > 0,\mathrm{Re}\,\beta > 0,a \in \mathbb{R},a\not =0\)
$$\displaystyle{ \left (I_{-}^{\alpha }\,t^{-\alpha -\gamma }E_{\beta,\gamma }(\mathit{at}^{-\beta })\right )(x) = x^{-\gamma }\left (E_{\beta,\alpha +\gamma }(\mathit{ax}^{-\beta })\right ). }$$
(4.10.11)
If we suppose additionally that Re (α +γ) > Re β, then the last formula can be rewritten as
$$\displaystyle{ \left (I_{-}^{\alpha }\,t^{-\alpha -\gamma }E_{\beta,\gamma }(\mathit{at}^{-\beta })\right )(x) = \frac{x^{\beta -\gamma }} {a} \left (E_{\beta,\alpha +\gamma -\beta }(\mathit{ax}^{-\beta }) - \frac{1} {\varGamma (\alpha +\gamma -\beta )}\right ), }$$
(4.10.12)
and, in particular,
$$\displaystyle{ \left (I_{-}^{\alpha }\,t^{-\alpha -\beta }E_{\alpha,\beta }(\mathit{at}^{-\alpha })\right )(x) = \frac{x^{\alpha -\beta }} {a} \left (E_{\alpha,\beta }(\mathit{ax}^{-\alpha }) - \frac{1} {\varGamma (\beta )}\right ). }$$
(4.10.13)
In the case of the fractional differentiation of the two-parametric Mittag-Leffler function we have
$$\displaystyle{ \left (D_{0+}^{\alpha }\,t^{\gamma -1}E_{\beta,\gamma }(\mathit{at}^{\beta })\right )(x) = x^{\gamma -\alpha -1}\left (E_{\beta,\gamma -\alpha }(\mathit{ax}^{\beta })\right ),\;\mathrm{Re}\,\alpha > 0,\;\mathrm{Re}\,\beta > 0,a \in \mathbb{R}. }$$
(4.10.14)
If we assume extra conditions on the parameters, namely Re γ > Re β, Re γ > Re (α +β), a ≠ 0, then the following relations hold:
$$\displaystyle{ \left (D_{0+}^{\alpha }\,t^{\gamma -1}E_{\beta,\gamma }(\mathit{at}^{\beta })\right )(x) = \frac{x^{\gamma -\alpha -\beta -1}} {a} \left (E_{\beta,\gamma -\alpha -\beta }(\mathit{ax}^{\beta }) - \frac{1} {\varGamma (\gamma -\alpha -\beta )}\right ). }$$
(4.10.15)
In particular (see [KilSai95b]), for Re α > 0,  Re β > Re α + 1, one can prove
$$\displaystyle{ \left (D_{0+}^{\alpha }\,t^{\beta -1}E_{\alpha,\beta }(\mathit{at}^{\alpha })\right )(x) = \frac{x^{\beta -\alpha -1}} {\varGamma (\beta -\alpha )} + \mathit{ax}^{\beta -1}\left (E_{\alpha,\beta }(\mathit{ax}^{\alpha })\right ). }$$
(4.10.16)
Finally, the right-sided (Liouville) fractional derivative of the two-parametric Mittag-Leffler function satisfies the relation (see, e.g., [KiSrTr06, p. 86])
$$\displaystyle{ \left (D_{-}^{\alpha }\,t^{\alpha -\beta }E_{\alpha,\beta }(\mathit{at}^{-\alpha })\right )(x) = \frac{x^{-\beta }} {\varGamma (\beta -\alpha )} + \mathit{ax}^{-\alpha -\beta }\left (E_{\alpha,\beta }(\mathit{ax}^{-\alpha })\right ), }$$
(4.10.17)
valid for all Re α > 0,  Re β > Re α + 1.

We also mention two extra integral relations for the two-parametric Mittag-Leffler function which are useful for applications.

Lemma 4.27.

Let α > 0 and β > 0. Then the following formula is valid
$$\displaystyle{ \frac{1} {\varGamma (\alpha )}\int \limits _{0}^{x}\frac{t^{\beta -1}E_{ 2\alpha,\beta }(t^{2\alpha })} {(x - t)^{1-\alpha }} \mathrm{d}t = x^{\beta -1}\left [E_{\alpha,\beta }(x^{\alpha }) - E_{ 2\alpha,\beta }(x^{2\alpha })\right ]. }$$
(4.10.18)

Corollary 4.28.

Formula(4.10.18)means
$$\displaystyle{ I_{0+}^{\alpha }\left (t^{\beta -1}E_{ 2\alpha,\beta }(t^{2\alpha })\right )(x) = x^{\beta -1}\left [E_{\alpha,\beta }(x^{\alpha }) - E_{ 2\alpha,\beta }(x^{2\alpha })\right ]. }$$
(4.10.19)

4.11 Historical and Bibliographical Notes

The two-parametric Mittag-Leffler function first appeared in the paper by Wiman 1905 [Wim05a], but he did not pay too much attention to it. Much later this function was rediscovered by Humbert and Agarval, who studied it in detail in 1953 [Aga53] (see also [Hum53, HumAga53]). A new function was obtained by replacing the additive constant 1 in the argument of the Gamma function in ( 3.1.1) by an arbitrary complex parameter β. Later, when we deal with Laplace transform pairs, the parameter β will be required to be positive like α.

Using the integral representations for E α, β (z) Dzherbashian [Dzh54a], [Dzh54b], [Dzh66, Ch. III, §2] proved formulas for the asymptotic representation of E α, β (z) at infinity, and in [Dzh66, Ch. III, §4] he gave applications of these to the construction of Fourier type integrals and to the proof of theorems on pointwise convergence of these integrals on functions defined and summable with exponential-power weight on a finite system of rays. Note that the developed technique is based on the representation of entire functions in the form of sums of integral transforms with kernels of the form E α, β (z).

By using asymptotic properties of the function E α, β (z), Dzherbashian (see[Dzh66, Ch. III, §2]) found its Mellin transform, established certain functional identities and proved the inversion formula for the following integral transform with the function E α, β (z) in the kernel
$$\displaystyle{ \int \limits _{0}^{\infty }E_{\alpha,\beta }(\mathrm{e}^{i\varphi }x^{\alpha }t^{\alpha })t^{\beta -1}f(t)\mathrm{d}t }$$
(4.11.1)
in the space \(L_{2}(\mathbb{R}_{+})\).
In [Bon-et-al02] the properties of the integral transforms with Mittag-Leffler function in the kernel
$$\displaystyle{ \int \limits _{0}^{\infty }E_{\alpha,\,\beta }(-\mathit{xt})f(t)\mathrm{d}t\ \ (x > 0) }$$
(4.11.2)
are studied in weighted spaces of r-summable functions
$$\displaystyle{ \mathcal{L}_{\nu,r} = \left \{f:\| f\|_{\nu,r} \equiv \left (\int \limits _{0}^{\infty }\vert t^{\nu }f(t)\vert \,\frac{\mathrm{d}t} {t} \right )^{1/r},\;\;\;1 \leq r < \infty,\;\;\;\nu \in \mathbb{R}\right \}. }$$
(4.11.3)
The conditions for the boundedness of such an operator as a mapping from one space to another were found, the images of these spaces under such a mapping were described, and inversion formulas were established. These results are based on the representation of (4.11.2) as a special case of the general H-transform (see Sect. F.3).

In recent years mathematicians’ attention towards the Mittag-Leffler type functions has increased, both from the analytical and numerical point of view, overall because of their relation to the fractional calculus. In addition to the books and papers already quoted in the text, here we would like to draw the reader’s attention to some recent papers on the Mittag-Leffler type functions, e.g., Al Saqabi and Tuan [Al-STua96], Kilbas and Saigo [KilSai96], Gorenflo, Luchko and Rogosin [GoLuRo97] and Mainardi and Gorenflo [MaiGor00]. Since the fractional calculus has now received wide interest for its applications in different areas of physics and engineering, we expect that the Mittag-Leffler function will soon occupy its place as the Queen Function of Fractional Calculus.

The remarkable asymptotic properties of the Mittag-Leffler function have provoked an interest in the investigation of the distribution of the zeros of E α, β (z). Several articles have been devoted to this problem (see [Dzh84, DzhNer68, OstPer97, Poly21, Pop02, Psk05, Sed94, Sed00, Wim05b]). An extended survey of the results is presented in [PopSed11]. Also studied is the related question of the distribution of zeros of sections and tails of the Mittag-Leffler function (see [Ost01, Zhe02]) and of some associated special functions (see [GraCso06, Luc00]).

The obtained results have found an application in the study of certain problems in spectral theory (see, e.g. [Dzh70, Djr93, Nak03]), approximation theory (see, e.g. [Sed98]), and in treating inverse problems (see, e.g., [TikEid02]).

Except in the case when \(\alpha = 1,\beta = -m,m \in \{-1\} \cup \mathbb{Z}_{+}\), the function E α, β (z) has an infinite set of zeros (see [Sed94]). In [Wim05b] it was shown that for α ≥ 2 all zeros of the classical Mittag-Leffler function E α, 1(z) are negative and simple (see also [Poly21], where the case \(\alpha = N \in \mathbb{N},N > 1\), is considered). In [Dzh84] it was proved that same result is valid for E 2, β (z), 1 < β < 3. Note that all zeros of the function \(E_{2,3}(z) =\cosh \frac{\sqrt{z}-1} {z}\) are twofold and negative, but the function E 2, β (z), β > 3, has no real zero.

Ostrovski and Pereselkova [OstPer97] formulated the problem to describe the set \(\mathcal{W}\) of pairs (α, β) such that all zeros of E α, β (z) are negative and simple. The authors conjectured that
$$\displaystyle{\mathcal{W} =\{ (\alpha,\beta )\vert \alpha \geq 2,0 <\beta < 1 +\alpha \}.}$$
It was shown, in particular, that \((\alpha,1),(\alpha,2) \in \mathcal{W}\) for all real α ≥ 2, and \(\left \{(2^{m},\beta )\vert m \in \mathbb{N},0 <\beta < 1 + 2^{m}\right \} \subset \mathcal{W}\).
The asymptotic behaviour of the zeros of the function E α, β (z) is the subject of several investigations. In [Sed94] asymptotic formulas for the zeros z n (α, β) of E α, β (z) were found for all α > 0 and \(\beta \in \mathbb{C}\). For 0 < α < 2 this asymptotic representation as n → ± is more exact and has the form
$$\displaystyle\begin{array}{rcl} & \left (z_{n}(\alpha,\beta )\right )^{1/\alpha } = 2\pi \mathit{in} + a(\alpha,\beta )\left (\log \,\vert n\vert + \frac{\pi i} {2}\mathrm{sign}\,n\right )& {}\\ & +b(\alpha,\beta ) + O\left (n^{-\alpha } + \frac{1} {n}\log \,\vert n\vert \right ). & {}\\ \end{array}$$
The values of a(α, β), b(α, β) are given in [Sed94]. A way of evaluating the zeros compatible with this asymptotical formula is proposed in [Sed00].

Schneider [Sch96] has proved that the generalized Mittag-Leffler function E α, β (−x) is completely monotonic for positive values of parameters α, β if and only if 0 < α ≤ 1, β ≥ α. The proof was based on the use of the corresponding probability measures and the Hankel integration path.

An analytic proof presented in Sect. 4.1.5 is due to Miller and Samko [MilSam97]. Note that the main formula (4.10.7) used in this proof is a special case of a more general relation due to Dzherbashian [Dzh66, p. 120] which states that \(x^{\beta +\gamma -1}E_{\alpha,\beta +\gamma }(-x^{\alpha })\) is the fractional integral of order γ of the function \(x^{\beta -1}E_{\alpha,\beta }(-x^{\alpha })\). However, the above presented result is more simple and straightforward.

Later (see, [MilSam01]), the proof of the complete monotonicity of some other special functions was given by Miller and Samko.

As a challenging open problem related to the Special Functions of Fractional Calculus (such as the multi-index Mittag-Leffler functions), we mention the possibility of their numerical computation and graphical interpretation, plots and tables, and implementations in software packages such as Mathematica, Maple, Matlab, etc. As mentioned earlier, the Classical Special Functions are already implemented there. For their Fractional Calculus analogues, numerical algorithms and software packages have been developed only for the classical Mittag-Leffler function E α; β (z) and the Wright function ϕ(α, β; z)! Numerical results and plots for the Mittag-Leffler functions for basic values of indices can be found in Caputo–Mainardi [CapMai71b] (one of the first attempts!) and Gorenflo–Mainardi [GorMai97]. Among the very recent achievements, we mention the following results: Podlubny [Pod11] (a Matlab routine that calculates the Mittag-Leffler function with desired accuracy), Gorenflo et al. [GoLoLu02] and Diethelm et al. [Die-et-al05] (algorithms for the numerical evaluation of the Mittag-Leffler function and a package for computation with Mathematica), Hilfer–Seybold [HilSey06] (an algorithm for extensive numerical calculations for the Mittag-Leffler function in the whole complex plane, based on its integral representations and exponential asymptotics), Luchko [Luc08] (algorithms for computation of the Wright function with prescribed accuracy), etc.

The results concerning calculation of the Mittag-Leffler function presented in Sect. 4.7 are based on the paper [GoLoLu02]. In [GoLoLu02] a numerical scheme for computation of the Mittag-Leffler function is given in pseudocode using a specially developed algorithm based on the above formulated results (see also the MatLab routine by Podlubny [Pod06, Pod11], and numerical computations of the Mittag-Leffler function performed by Hilfer and Seybold [SeyHil05, HilSey06, SeyHil08]).

The increasing interest in Mittag-Leffler functions and their wide application has given rise to the need for effective strategies for their numerical computation (see [HilSey06, Pod11, SeyHil08]). Thus the analysis of efficient and accurate numerical methods for generalized Mittag-Leffler functions with two parameters
$$\displaystyle{e_{\alpha,\beta }(t;\lambda ) = t^{\beta -1}E_{\alpha,\beta }(-\lambda t^{\alpha }),}$$
with scalar or matrix arguments, is actually a compelling subject. In [GarPop12] the problem of the numerical computation of generalized Mittag-Leffler functions with two parameters e α, β (t; λ) is studied with applications to fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are used and the quadrature rules for numerical integration are applied. An in-depth error analysis is carried out to select suitable contour parameters, depending on the parameters of the Mittag-Leffler function, in order to achieve any fixed accuracy. Numerical experiments to validate theoretical results are performed and some computational issues are discussed.

4.12 Exercises

  1. 4.12.1.
    Prove the following relations ([Ber-S05b]):
    $$\displaystyle\begin{array}{rcl} E_{\alpha }(-x) = E_{2\alpha }(x^{2}) -\mathit{xE}_{ 2\alpha,1+\alpha }(x^{2}),\;\;\;x \in \mathbb{R},\;\;\;\mathrm{Re}\,\alpha > 0,& & {}\end{array}$$
    (4.12.1a)
    $$\displaystyle\begin{array}{rcl} E_{\alpha }(-\mathit{ix}) = E_{2\alpha }(-x^{2}) -\mathit{ix}E_{ 2\alpha,1+\alpha }(-x^{2}),\;\;\;x \in \mathbb{R},\;\;\;\mathrm{Re}\,\alpha > 0.& & {}\end{array}$$
    (4.12.1b)
     
  2. 4.12.2.
    Prove the following recurrence relations ([SaKaKi03])
    $$\displaystyle{ z^{m}E_{\alpha,\beta +m\alpha }(z)=E_{\alpha,\beta }(z) -\sum \limits _{n=0}^{m-1} \frac{z^{n}} {\varGamma (\beta +n\alpha )},\;\;\;\mathrm{Re}\,\alpha >0,\;\mathrm{Re}\,\beta >0,\;m \in \mathbb{N}. }$$
    (4.12.2)
     
  3. 4.12.3.
    Let the family of functions H α be given by the formula ([Ber-S05b, p. 432])
    $$\displaystyle{H_{\alpha }(k) = \frac{2} {\pi } \int \limits _{0}^{\infty }E_{ 2\alpha }(-t^{2}) \cdot \cos \, (\mathit{kt})\mathrm{d}t,\;\;\;k > 0,\;0 \leq \alpha \leq 1,}$$
    where its power series in k has the form
    $$\displaystyle{H_{\alpha }(k) = \frac{1} {\pi } \sum \limits _{n=0}^{\infty }b_{ n}(\alpha )k^{n},\;\;\;0 \leq \alpha < 1.}$$
    Deduce the following asymptotic formula for E α (−x):
    $$\displaystyle{E_{\alpha }(-x) = \frac{1} {\pi } \sum \limits _{n=0}^{\infty } \frac{b_{n}(\alpha )} {x^{n+1}},\;\;\;0 \leq \alpha < 1.}$$

    Hint. Use the relation (4.12.1a).

     
  4. 4.12.4.
    Using the series representation of the two-parametric Mittag-Leffler function (4.1.1) prove the following recurrence relations
    1. 4.12.4.1.
      $$\displaystyle\begin{array}{rcl} & E_{1,1}(z) + E_{1,1}(-z) = 2\,E_{2,1}(z^{2})\,\Longleftrightarrow\,\mathrm{e}^{z} +\mathrm{ e}^{-z} = 2\,\cosh (z)\,, & {}\\ & E_{1,1}(z) - E_{1,1}(-z) = 2z\,E_{2,2}(z^{2})\,\Longleftrightarrow\,\mathrm{e}^{z} -\mathrm{ e}^{-z} = 2\,\sinh (z)\,,& {}\\ \end{array}$$
      or in more general form:
       
    2. 4.12.4.2.
      $$\displaystyle\begin{array}{rcl} & E_{\alpha,\beta }(z) + E_{\alpha,\beta }(-z) = 2\,E_{2\alpha,\beta }(z^{2})\,, & {}\\ & E_{\alpha,\beta }(z) - E_{\alpha,\beta }(-z) = 2z\,E_{2\alpha,\alpha +\beta }(z^{2})\,;& {}\\ \end{array}$$
       
    3. 4.12.4.3.
      $$\displaystyle{E_{1,3}(z) = \frac{\mathrm{e}^{z} - 1 - z} {z^{2}} }$$
      or in more general form (for any \(m \in \mathbb{N}\)):
       
    4. 4.12.4.4.
      $$\displaystyle{E_{1,m}(z) = \frac{1} {z^{m-1}}\left \{\mathrm{e}^{z} -\sum \limits _{ k=0}^{m-2}\frac{z^{k}} {k!} \right \}.}$$
       
     
  5. 4.12.5.
    With α > 0 show that
    $$\displaystyle{ t^{\alpha -1}\,E_{\alpha,\alpha }(-t^{\alpha }) = -\frac{\mbox{ d}} {\mbox{ d}t}E_{\alpha }(-t^{\alpha })\,. }$$
    (2par - 1par)
     
  6. 4.12.6.
    Prove the following differential relations for the two-parametric Mittag-Leffler function ([GupDeb07])
    1. 4.12.6.1.
      $$\displaystyle{3E_{1,4}(z) + 5zE_{1,4}^{{\prime}} + z^{2}E_{ 1,4}^{{\prime\prime}} = E_{ 1,2} - E_{1,3}.}$$
       
    2. 4.12.6.2.
      $$\displaystyle\begin{array}{rcl} & & n(n + 2)E_{\alpha,n+3}(z) + z\alpha [2n +\alpha +2]E_{\alpha,n+3}^{{\prime}} + z^{2}E_{\alpha,n+3}^{{\prime\prime}} {}\\ & =& E_{\alpha,n+1} - E_{\alpha,n+2}, {}\\ \end{array}$$
      which hold for any α > 0 and any n = 1, 2, .
       
     
  7. 4.12.7.
    Prove the following Laplace transform pair for the auxiliary functions of Mittag-Leffler type defined below
    $$\displaystyle{ e_{\alpha,\beta }(t;\lambda ):= t^{\beta -1}\,E_{\alpha,\beta }\left (-\lambda \,t^{\alpha }\right ) \div \frac{s^{\alpha -\beta }} {s^{\alpha }+\lambda } = \frac{s^{-\beta }} {1 +\lambda s^{-\alpha }}\,. }$$
    (LT - E2)
     
  8. 4.12.8.
    Evaluate the following integrals ([HaMaSa11]):
    1. 4.12.8.1.
      $$\displaystyle{\int \limits _{0}^{x} \frac{E_{\alpha }(t^{\alpha })} {(x - t)^{1-\beta }}\mathrm{d}t}$$
      for Re α > 0, Re β > 0.
       
    2. 4.12.8.2.
      $$\displaystyle{\int \limits _{0}^{\infty }\mathrm{e}^{-\mathit{st}}t^{m\alpha +\beta -1}E_{\alpha,\beta }^{(m)}(\pm \mathit{at}^{\alpha })\mathrm{d}t}$$
      for Re s > 0, Re α > 0, Re β > 0, where
      $$\displaystyle{E_{\alpha,\beta }^{(m)}(z) = \frac{\mathrm{d}^{m}} {\mathrm{d}z^{m}}E_{\alpha,\beta }(z).}$$
       

    Answers.

    1. 4.12.8.1.
      $$\displaystyle{\varGamma (\beta )x^{\beta }E_{\alpha,\beta +1}(x^{\alpha }).}$$
       
    2. 4.12.8.2.
      $$\displaystyle{ \frac{m!s^{\alpha -\beta }} {(s^{\alpha } \mp a)^{m+1}}.}$$
       
     
  9. 4.12.9.
    Prove the following formulas for half-integer values of parameters [Han-et-al09]:
    1. 4.12.9.1.
      $$\displaystyle{E_{1/2,1/2}(\pm x) = \frac{1} {\sqrt{x}} \pm x\mathrm{e}^{x^{2} }[1 \pm \mathrm{erf}(x)].}$$
       
    2. 4.12.9.2.
      $$\displaystyle{E_{1/2,1}(\pm x) =\mathrm{ e}^{x^{2} }[1 \pm \mathrm{erf}(x)].}$$
       
    3. 4.12.9.3.
      $$\displaystyle{E_{1,1/2}(+x) = \frac{1} {\sqrt{x}} + \sqrt{x}\mathrm{e}^{+x}\mathrm{erf}(\sqrt{x}).}$$
       
    4. 4.12.9.4.
      $$\displaystyle{E_{1,1/2}(-x) = \frac{1} {\sqrt{x}} + i\sqrt{x}\mathrm{e}^{-x}\mathrm{erf}(i\sqrt{x}).}$$
       
    5. 4.12.9.5.
      $$\displaystyle{E_{1,3/2}(+x) =\mathrm{ e}^{+x}\frac{\mathrm{erf}(\sqrt{x})} {\sqrt{x}}.}$$
       
    6. 4.12.9.6.
      $$\displaystyle{E_{1,3/2}(-x) = -i\mathrm{e}^{-x}\frac{\mathrm{erf}(i\sqrt{x})} {\sqrt{x}}.}$$
       
     
  10. 4.12.10.
    Prove the following formulas for negative integer values of parameters [Han-et-al09]:
    1. 4.12.10.1.
      $$\displaystyle{E_{-1,2}\left (\pm \frac{1} {x}\right ) = 1 \pm \frac{1 -\mathrm{ e}^{\pm x}} {x}.}$$
       
    2. 4.12.10.2.
      $$\displaystyle{E_{-2,1}\left (+ \frac{1} {x^{2}}\right ) = 1 -\cosh x.}$$
       
    3. 4.12.10.3.
      $$\displaystyle{E_{-2,1}\left (-\frac{1} {x^{2}}\right ) = 1 -\cos x.}$$
       
    4. 4.12.10.4.
      $$\displaystyle{E_{-2,2}\left (+ \frac{1} {x^{2}}\right ) = 1 -\frac{\sinh x} {x}.}$$
       
    5. 4.12.10.5.
      $$\displaystyle{E_{-2,2}\left (-\frac{1} {x^{2}}\right ) = 1 -\frac{\sin x} {x}.}$$
       
     
  11. 4.12.11.
    Prove the following formulas for negative semi-integer values of parameters [Han-et-al09]:
    1. 4.12.11.1.
      $$\displaystyle{E_{-1/2,1/2}\left (\pm \frac{1} {x}\right ) = \mp x\mathrm{e}^{x^{2} }[1 \pm \mathrm{erf}(x)].}$$
       
    2. 4.12.11.2.
      $$\displaystyle{E_{-1/2,1}\left (\pm \frac{1} {x}\right ) = 1 -\mathrm{ e}^{x^{2} }[1 \pm \mathrm{erf}(x)].}$$
       
    3. 4.12.11.3.
      $$\displaystyle{E_{-1,1/2}\left (+\frac{1} {x}\right ) = \sqrt{x}\mathrm{e}^{+x}\mathrm{erf}(\sqrt{x}).}$$
       
    4. 4.12.11.4.
      $$\displaystyle{E_{-1,1/2}\left (-\frac{1} {x}\right ) = -i\sqrt{x}\mathrm{e}^{-x}\mathrm{erf}(i\sqrt{x}).}$$
       
    5. 4.12.11.5.
      $$\displaystyle{E_{-1,3/2}\left (+\frac{1} {x}\right ) = \frac{2} {\sqrt{x}} -\mathrm{ e}^{+x}\frac{\mathrm{erf}(\sqrt{x})} {\sqrt{x}}.}$$
       
    6. 4.12.11.6.
      $$\displaystyle{E_{-1,3/2}\left (-\frac{1} {x}\right ) = \frac{2} {\sqrt{x}} + i\mathrm{e}^{-x}\frac{\mathrm{erf}(i\sqrt{x})} {\sqrt{x}}.}$$
       
     
  12. 4.12.12.
    Prove the following formula for the Laplace transform of the derivatives of the Mittag-Leffler function ([KiSrTr06, p. 50]):
    $$\displaystyle{\left (\mathcal{L}\,t^{\alpha n+\beta -1}\left (\frac{\partial } {\partial \lambda }\right )^{n}E_{\alpha,\beta }(\lambda t^{\alpha })\right )(s) = \frac{n!s^{\alpha -\beta }} {\left (s^{\alpha }-\lambda \right )^{n+1}}\;\;\;(\vert \lambda s^{-\alpha }\vert < 1).}$$
     
  13. 4.12.12.
    Prove the following relations for the Mittag-Leffler functions with positive integer values of parameters (Capelas relations) ([Cap13]).
    $$\displaystyle\begin{array}{rcl} \sum \limits _{k=1}^{m}z^{k-1}E_{ m,k}(z^{m}) =\mathrm{ e}^{z},\;\;\;m \in \mathbb{N},& & {}\end{array}$$
    (4.12.12a)
    $$\displaystyle\begin{array}{rcl} E_{1,m}(z) = \frac{1} {z^{m-1}}\left (\mathrm{e}^{z} -\sum \limits _{ k=0}^{m-2}\frac{z^{k}} {k!} \right ),\;\;\;m \in \mathbb{N}.& & {}\end{array}$$
    (4.12.12b)
     

Footnotes

  1. 1.

    The meaning of the weak limit \(\lim _{r\rightarrow \infty }^{\!\!\!{\ast}}\) is the same as in the definition of entire functions of completely regular growth, see (4.5.1).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Rudolf Gorenflo
    • 1
  • Anatoly A. Kilbas
    • 2
  • Francesco Mainardi
    • 3
  • Sergei V. Rogosin
    • 4
  1. 1.Free University Berlin Mathematical InstituteBerlinGermany
  2. 2.Belarusian State University Department of Mathematics and MechanicsMinskBelarus
  3. 3.University of Bologna Department of PhysicsBolognaItaly
  4. 4.Belarusian State University Department of EconomicsMinskBelarus

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