Abstract
The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann–Liouville fractional integration operator has been obtained.
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Background
The well-known Mittag-Leffler function \(E_{\alpha } (z)\) named after its originator, the Swedish mathematician Gosta Mittag-Leffler (1846–1927), is defined by (Mittag-Leffler 1903)
The Mittag–Leffler function naturally occurs as the solution of fractional order differential equations. The various generalization of Mittag–Leffler function have been defined and studied by different authors.
Shukla and Prajapati (2007) introduced its generalization \(E_{\alpha ,\beta }^{\gamma ,q} (z)\), this is defined as
for \(\alpha ,\beta ,\gamma \in C\); \(\text{Re} \left( \alpha \right) > 0,\,\text{Re} \left( \beta \right) > 0,\text{Re} \left( \gamma \right) > 0,\text{Re} \left( \delta \right) > 0\), \(q \in (0,1) \cup N\), and \((\gamma )_{qn} = \frac{\Gamma (\gamma + qn)}{\Gamma (\gamma )}\) denotes the generalized Pochhammer symbol.
Further, the generalization of (2) is also given by Khan and Ahmed (2013), as follows:
where \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon \in C\); \(p,q,\rho ,\sigma > 0\);\(q \le Re(\alpha ) + p\); \(\rho \le \text{Re} \left( \sigma \right) + p\); \(q \le \text{Re} \left( \sigma \right) + p\); \(\rho ,q \in (0,1) \cup N\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right), \, \text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right)} \right) > 0.\)
Here, the convergence conditions of (3) have been modified, which was given by Khan and Ahmed (2013).
The following well-known notations and definitions have been used:
Let \(L(a,b)\) (Kilbas et al. 2004) be a set of all Lebesgue measurable real or complex valued functions \(f(x)\) on \([a,b]\) i.e.
Let \(f(x) \in L(a,b)\), \(\mu \in C\) \((\text{Re} (\mu ) > 0)\) then the Riemann–Liouville left-sided fractional integrals of order \(\mu\) (Miller and Ross 1993) is defined as
and the R–L right-sided fractional integral of order \(\mu\) is defined as
Miller and Ross (1993) defined the following:
If \(\mu ,\alpha ,\beta \in C\), \(\text{Re} (\mu ) > 0\); \(n = [\text{Re} (\mu )] + 1\); \({\text{Re}} (\beta ) > 0\) then
and
Khan and Ahmed (2013) proved the following result.
If \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\); \(q \le Re(\alpha ) + p\) and
then for \(m \in N,\)
In continuation of study, in this paper we give the operator associated with \(E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (z)\) as follows:
Let \(f(x) \in L(a,b)\), define
where \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon \in C\); \(p,q,\rho ,\sigma > 0\); \(q \le Re(\alpha ) + p\); \(\rho \le \text{Re} \left( \sigma \right) + p\); \(q \le \text{Re} \left( \sigma \right) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right)} \right) > 0.\)
Main results
Using the definition (4), one can easily prove following lemma.
Lemma 1
If \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\); \(q \le Re(\alpha ) + p\); \(x > a\); \(a \in R_{ + } = [0,\infty )\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\) then
Theorem 1
Let \(a \in R_{ + } = [0,\infty )\). Let \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\) ; \(q \le Re(\alpha ) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,x > a.\) Then
Proof
Using definitions (3) and (5) and further simplification gives
This completes the proof of (12).□
To prove (13), we use definitions (8) and further simplification gives
On applying (12) with replacement of \(r\) by \(n - r\), the above equation reduces to
From (9), we get
Theorem 2
Let \(a \in R_{ + } = [0,\infty ),\) \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\) and \(q \le Re(\alpha ) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0\) \(x > a.\) Then
Proof
Taking \(f(t) = (t - a)^{r - 1}\) in (10), we get
Replacing \(t\) by \(a + (x - a)t\) and simplifying the above equation
and further simplification of above equation gives the proof of Theorem 2.
Theorem 3
Let \(a \in R_{ + } = [0,\infty ),\) \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\) and \(q \le Re(\alpha ) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\) \(b > a.\), Then the operator \(E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q}\) is bounded on \(L(a,b)\) and
where
Proof
On using the definition (10) and applying Dirichlet’s formula (Samko et al. 1993), we have
Taking \(u = x - t\) in inner integral, this yields
This completes the proof.□
Theorem 4
(Composition with Riemann–Liouville fractional integration operator) Let \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\); \(q \le Re(\alpha ) + p\); \(b > a\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0.\) Then the relation
holds for any summable function \(f \in L(a,b).\)
Proof
Applying Dirichlet’s formula (Samko et al. 1993), we get
Substituting \(u - t = \tau\) in the above equation, we get
Again using (5), this equation becomes
Applying (12), this yields
Using (10), we get
The other equality can also be proved in the similar way.
Theorem 5
Let \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\); \(q \le Re(\alpha ) + p\); \(b > a\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0.\) Then the relation
holds for any continuous function \(f \in C[a,b]\).
Proof
From (8), we have
Again using Theorem 4 and definition (10),
The integrand in the above equation is continuous function on \([a,b]\), here we take
Applying same procedures as above, this led the proof of the theorem. This is easy to prove by using mathematical induction method also.
Theorem 6
Let \(a \in R_{ + } = [0,\infty ),\) \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0;\) \(q \le Re(\alpha ) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,\,x > a.\) Then
Proof
We have
This completes the proof.□
Corollary 1
If \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\) and \(q \le Re(\alpha ) + p\) \(x > a\); \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,\,a \in R_{ + } = [0,\infty ).\) Let \(I_{0 + }^{r}\) be the left-sided operator of Riemann–Liouville fractional integral. Then
Proof is very obvious from Lemma 1 and Theorem 6.
Theorem 7
Let \(a \in R_{ + } = [0,\infty )\), \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0;\) \(q \le Re(\alpha ) + p\) and \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,x > a,\) \(I_{ - }^{r}\) be the right-sided operator of Riemann–Liouville fractional integral. Then
Proof
Let
On changing the order of the summation and integration then afterward applying beta function, this gives
Corollary 2
If \(\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C\); \(p,q > 0\) and \(q \le Re(\alpha ) + p; \,\,\, x > a\) ; \(\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\) \(a \in R_{ + } = [0,\infty )\). Let \(I_{ - }^{r}\) be the right-sided operator of Riemann–Liouville fractional integral. Then
Conclusion
In this paper, we proved some properties of generalized Mittag-Leffler functions and also used the fractional calculus approach to prove Theorems 4, 5, 6 and 7.
References
Khan MA, Ahmed S (2013) On some properties of the generalized Mittag-Leffler function. SpringerPlus 2:1–9
Kilbas AA, Sagio M, Saxena RK (2004) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Trans Spec Funct 15(1):31–49
Miller KS, Ross B (1993) An introduction to fractional calculus and fractional differential equations. Wiley, New York
Mittag- Leffler G (1903) Sur la nouvelle function E α(x). C R Acad Sci Paris 137:554–558
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, theory and applications. Gordon and Breach, New York
Shukla AK, Prajapati JC (2007) On a generalization of Mittag-Leffler function and its properties. J Math Anal Appl 336(2):797–811
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Desai, R., Salehbhai, I.A. & Shukla, A.K. Note on generalized Mittag-Leffler function. SpringerPlus 5, 683 (2016). https://doi.org/10.1186/s40064-016-2299-x
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DOI: https://doi.org/10.1186/s40064-016-2299-x