1 Introduction, Earlier Results and Motivation

Let \(X_{1}, X_{2}, \dots , X_{\nu }\) are independent homoscedastic normally distributed random variables (rv)s with means \(\mu _{j}, j=\overline{1, \nu }\) and the same (unit) variance defined on a standard probability space \((\Omega , {\mathscr {F}}, {\mathsf {P}})\). Then the rv \(\Xi = X_{1}^{2}+ \dots + X_{\nu }^{2}\) has non-central \(\chi ^{2}\) distribution of \(\nu \in {\mathbb {N}}\) degrees of freedom (the size number of the sum), with the non-centrality parameter \(\lambda = \mu _{1}^{2} + \cdots +\mu _{\nu }^{2} \ge 0\) [9,10,11]. The distribution of the rv \(\Xi \) is usually denoted by \(\chi _{\nu }^{\prime 2}(\lambda )\) [11, p. 433]; in this case we write \(\Xi \sim \chi _{\nu }^{\prime 2}(\lambda )\). The associated probability density function (PDF) can be represented [7, 11] in terms of the modified Bessel function of the first kind \(I_{\eta }\) with the order \(\eta \):

$$\begin{aligned} f_{\nu ,\lambda }(x) = \dfrac{1}{2}\, \mathrm{e}^{-\frac{x+\lambda }{2}} \left( \dfrac{x}{\lambda }\right) ^{\frac{\nu -2}{4}} \,I_{\frac{\nu }{2}-1}(\sqrt{\lambda x}\,), \qquad x>0. \end{aligned}$$

Due to the usefulness and wide application of the non-central \(\chi ^{2}\)-distribution, e.g. in finance, estimation and decision theory, time series analysis [1, 21], mathematical physics [11, p. 435], certain statistical tests [12], etc., many authors have studied the correspoding PDF and cumulative distribution function (CDF), mostly in the case when \(\nu = n \in {\mathbb {N}}\) (see e.g. [3, 11, 17, 18, 20, 23]). Lately, the case \(\nu = 2n \in {\mathbb {N}}\) is strongly considered by the authors in [2, 9, 10], even though, in general, \(\nu \) can be a nonnegative real number [19].

It is also important to emphasize that \(\chi _{\nu }^{\prime 2}(\lambda )\) also has its applications in communication theory, when the corresponding CDF is given by [9, p. 66, Eq. (1.1)]

$$\begin{aligned} F_{\nu ,\lambda }(x) = 1-Q_{\frac{\nu }{2}}(\sqrt{\lambda },\sqrt{x}\,),\qquad x>0, \end{aligned}$$
(1)

where \(Q_{\mu }(a, b)\) is the generalized Marcum Q function [15]

$$\begin{aligned} Q_{\mu }(a,b) = \dfrac{1}{a^{\mu -1}}\int _{b}^{\infty }t^{\mu } \mathrm{e}^{-\frac{t^{2}+a^{2}}{2}}\, I_{\mu -1}(at)\, \mathrm{d}t, \qquad a,\mu >0,\,b \ge 0; \end{aligned}$$
(2)

in this case the non-centrality parameter \(\lambda \) is interpreted as a signal-to-noise ratio [11, p. 435].

Using the representation (1), Jankov Maširević showed that the CDF of the noncentral \(\chi ^{2}\) distribution, in the case of even number of the degrees of freedom, can be represented in the following form [9, p. 4, proof of Theorem 2.1)]:

$$\begin{aligned} F_{2n,\lambda }(x) = Q_{1}(\sqrt{x},\sqrt{\lambda }) - \mathrm{e}^{-\frac{\lambda +x}{2}}\sum _{m=1}^{n} \left( \sqrt{\dfrac{x}{\lambda }}\right) ^{m-1}I_{1-m} (\sqrt{\lambda x}), \end{aligned}$$
(3)

where \(n\in {\mathbb {N}}\) and \(\lambda , x >0\).

The literature covering \(Q_{1}\) contains several recent and up-to-date references and closed form results. For instance, Brychkov derived the closed-form formula for the generalized Marcum Q function of general non-negative integer order [3, p. 178, Eq. (10)]

$$\begin{aligned} Q_{n+1}(a,b) = \mathrm{e}^{-\frac{1}{2}(a^{2}+b^{2})}\,\Big \{ \Phi _{3}\Big ( 1; 1; \frac{a^{2}}{2}, \frac{a^{2} b^{2}}{4}\Big ) + \sum _{k=1}^{n} \Big (\frac{b}{a}\Big )^{k} \, I_{k}(a b)\Big \}. \end{aligned}$$

The special case of this formula for \(n = 0\) appears in [3, p. 179, Eq. (12)] as

$$\begin{aligned} Q_{1}(a,b) = \mathrm{e}^{-\frac{a^{2}+b^{2}}{2}}\,\Phi _3\Big (1;1;\dfrac{a^{2}}{2},\dfrac{a^{2}b^{2}}{4} \Big ), \end{aligned}$$
(4)

the empty sum is understood to be equal to zero and the Humbert function \(\Phi _{3}\) is defined by the series [6, p. 225, Eq. (22)]

$$\begin{aligned} \Phi _{3}\big (\alpha ; \beta ; z, w\big ) = \sum _{k=0}^\infty \sum _{m=0}^\infty \frac{(\alpha )_k}{(\beta )_{k+m}} \frac{z^k}{k!} \frac{w^m}{m!}, \qquad \max \{|z|, |w|\} < \infty . \end{aligned}$$

Sofotasios et al. [22, p. 7811, Eq. (109)] proved the relation in terms of the equal, unit-parameter Humbert \(\Phi _1\) function, reads as follows

$$\begin{aligned} Q_1(b,c)- Q_1(c,b) = 1 - \mathrm{e}^{-\frac{y}{x}} \sqrt{1-x}\, \Phi _1\Big (\frac{1}{2},1;1;x,y\Big ), \quad y>0,\,|x|<1, \end{aligned}$$
(5)

and

$$\begin{aligned} b = \sqrt{\frac{y}{x}(1+\sqrt{1-x})-\frac{y}{2}}, \qquad c = \sqrt{\frac{y}{x}(1-\sqrt{1-x})-\frac{y}{2}};\end{aligned}$$

where the Humbert function \(\Phi _1\big (\alpha , \beta ; \gamma ; z, w\big )\) is defined by the relation [6, p. 225, Eq. (20)]

$$\begin{aligned} \Phi _{1}\big (\alpha , \beta ; \gamma ; w, z\big ) = \sum _{k=0}^\infty \sum _{m=0}^\infty \dfrac{(\alpha )_{k+m}(\beta )_k}{(\gamma )_{k+m}} \frac{z^k}{k!} \frac{w^m}{m!}, \qquad |w|<1. \end{aligned}$$

In fact, this is the abbreviated form of the relation [22, p. 7811, Eq. (109)], since \({}_{2}F_{1}\big (\tfrac{1}{2}, 1; 1; x\big ) = (1-x)^{-\frac{1}{2}}\).

Finally, it is worth to mention the Brychkov–Savischenko’s \({\mathcal {H}}_{p}\) function, which appears in evaluation of the error probability for generalized Rice–Nakagami fading conditions with coherent reception and relates to Marcum \(Q_{1}\), see [4].

Regarding the expression (3), the CDF of the rv \(\Xi \sim \chi _{\nu }^{\prime 2}(\lambda )\) admits representation formulae in terms of incomplete Lipschitz–Hankel \(I_{e_{\mu , \nu }}\) [16], and Rice \(I_{e}\) integrals [22], Temme’s T-integral [23], and also as the related S-integral, introduced by Jankov-Maširević [9], which have computational adventages. Also, we point out that another expressions for the CDF were obtained by completely different approach in the recent articles [2, 10].

The main aim of this paper is to derive simple and elegant expression for \(F_{2n,\lambda }\), relying on the formula (3) and some results considered by Goldstein [8, 14], which will be presented in the next section. The formula involves the Humbert hypergeometric series \(\Phi _{1}\) of two variables, Marcum function of the first order \(Q_{1}\) and a finite sum of modified Bessel functions of the first kind. Finally, it is worth to mention that as by-products, novel closed form expressions are obtained for the first order Marcum function \(Q_{1}\) and transformation formulae for Humbert \(\Phi _{1}\) and \(\Phi _{3}\). We also point out that only the Humbert \(\Phi _1\) possesses Euler-type single integral representation on the real axis with elementary functions in the integrand, which ensures a simpler approximation procedure in numerical evaluations using quadratures.

2 On CDF of \(\chi _{2n}^{\prime 2}(\lambda )\)-Distribution Regarding Goldstein’s Result

In this section we derive new representation formula for \(F_{2n,\lambda }\), presented in a closed form. In 1953 Goldstein [8, p. 158 et seq., §3.] has considered the following integral involving the modified Bessel function

$$\begin{aligned} J(u,v) = 1 - \mathrm{e}^{-v} \int _0^u \mathrm{e}^{-t}I_{0}\left( 2\sqrt{v t}\right) \,\mathrm{d}t,\end{aligned}$$

which appears in a variety of applied problems, see also Luke [14, Chapter XII].

The Marcum function \(Q_{1}\) of the first order is the special case of (2) for \(\mu =1\), which is

$$\begin{aligned} Q_{1}(\sqrt{x},\sqrt{\lambda })&= \mathrm{e}^{-\frac{x}{2}} \int _{\sqrt{\lambda }}^{\infty }t \mathrm{e}^{-\frac{t^{2}}{2}}\, I_{0}(\sqrt{x}\,t)\,\mathrm{d}t \nonumber \\&= 1- \mathrm{e}^{-\frac{x}{2}} \int _{0}^{\sqrt{\lambda }} t \mathrm{e}^{-\frac{t^{2}}{2}}\,I_{0}(\sqrt{x}\,t)\,\mathrm{d}t = J\left( \frac{\lambda }{2},\frac{x}{2}\right) . \end{aligned}$$
(6)

Thus, our aim is to find a closed formula for the function J(u, v) in order to get a new expression for CDF (3) in terms of \(Q_{1}\).

Now, we report Goldstein’s function [8, p. 162, Eq. (60)] in a slightly different form, i.e. transformed linearly from the original formula by the substitution \(t\rightarrow 4t\):

$$\begin{aligned} J(u,v) = {\left\{ \begin{array}{ll} \mathrm{e}^{-(u+v)}\,I(\xi ,\eta ), &{}\quad \eta <1\\ 1+\mathrm{e}^{-(u+v)}\,I(\xi ,\eta ), &{}\quad \eta >1 \end{array}\right. }, \end{aligned}$$
(7)

where \(\xi = 2\sqrt{u v}\) and \(\eta =\sqrt{v/u}\), while \(J(u, u) = \tfrac{1}{2}\,\big [1+ \mathrm{e}^{-2 u} I_{0}(2 u)\big ]\); compare [8, p. 160, Eq. (48)].

In terms of the auxiliary function [8, p. 162, Eq. (64)]

$$\begin{aligned} I(\xi ,\eta ) = \frac{1}{2}\, \Bigg \{ I_{0}(\xi )+\dfrac{(1-\eta ^{2}) \mathrm{e}^{\xi }}{\pi }\int _{0}^{1} \dfrac{\mathrm{e}^{-2 \xi \,t}\,\mathrm{d}t}{\sqrt{t-t^{2}}\,[(1-\eta )^{2}+4 \eta t]} \Bigg \}, \end{aligned}$$
(8)

and (7) we conclude that

$$\begin{aligned} Q_{1}(\sqrt{x},\sqrt{\lambda }) = {\left\{ \begin{array}{ll} \mathrm{e}^{-\frac{\lambda +x}{2}}\,I\big (\sqrt{\lambda x},\sqrt{\frac{x}{\lambda }}\big ), &{}\quad x <\lambda \\ 1 + \mathrm{e}^{-\frac{\lambda +x}{2}}\, I\big (\sqrt{\lambda x},\sqrt{\frac{x}{\lambda }}\big ), &{}\quad x >\lambda \end{array}\right. }, \end{aligned}$$

where

$$\begin{aligned} I\Big (\sqrt{\lambda x},\sqrt{\frac{x}{\lambda }}\Big ) = \frac{1}{2}\,\Bigg [ I_{0}(\sqrt{\lambda x}) + \dfrac{\lambda -x}{\pi }\, \mathrm{e}^{\sqrt{\lambda x}} \int _{0}^{1} \dfrac{(t-t^{2})^{-\frac{1}{2}}\, \mathrm{e}^{-2 \sqrt{\lambda x}\,t}\, \mathrm{d}t}{\big (\sqrt{\lambda }-\sqrt{x}\big )^{2} + 4\sqrt{\lambda x}\,t} \Bigg ]. \end{aligned}$$
(9)

Unlike Goldstein’s \(\xi , \eta \)-labeling, here and in what follows, we apply the new parametrization \(\xi ^{\prime }= \sqrt{\lambda x}\) and \(\eta ^{\prime }= \sqrt{x/\lambda }\); compare the arguments of the function J(u, v) in (6) and (7).

Now we are ready to formulate one of the main results, which is the explicit expression for the CDF of the \(\chi _{\nu }^{\prime 2}(\lambda )\) distribution with even degrees of freedom in terms of exponential function, modified Bessel function of the first kind of zero order and Humbert hypergeometric \(\Phi _{1}\) function of two variables.

Theorem 1

For all \(\min \{\lambda , x\}>0\) such that \(q_{4} = 4\sqrt{\lambda x} (\sqrt{\lambda }- \sqrt{x}\,)^{-2} <1\) and any \(n \in {\mathbb {N}}\)

$$\begin{aligned} F_{2n,\lambda }(x) = Q_{1}(\sqrt{x},\sqrt{\lambda }) - \mathrm{e}^{-\frac{\lambda +x}{2}}\sum _{m=1}^{n} \left( \sqrt{\dfrac{x}{\lambda }}\right) ^{m-1} I_{1-m}(\sqrt{\lambda x}), \end{aligned}$$
(10)

where the Marcum function has the relation

$$\begin{aligned} Q_{1}(\sqrt{x},\sqrt{\lambda }) = {\left\{ \begin{array}{ll} \tfrac{1}{2} \mathrm{e}^{-\frac{\lambda +x}{2}}\,\Big [I_{0}(\sqrt{\lambda x})+\tfrac{\sqrt{\lambda }+\sqrt{x}}{\sqrt{\lambda }-\sqrt{x}}\, \mathrm{e}^{\sqrt{\lambda x}} &{} \\ \qquad \times \Phi _{1}\Big ( \frac{1}{2}, 1; 1; \frac{-4\,\sqrt{\lambda x}}{(\sqrt{\lambda }-\sqrt{x}\,)^{2}}, -2 \sqrt{\lambda x}\Big )\Big ], &{}\quad x<\lambda \\ 1 + \tfrac{1}{2} \mathrm{e}^{-\frac{\lambda +x}{2}}\,\Big [I_{0}(\sqrt{\lambda x}\,) + \tfrac{\sqrt{\lambda }+\sqrt{x}}{\sqrt{\lambda }-\sqrt{x}}\, \mathrm{e}^{\sqrt{\lambda x}} &{} \\ \qquad \times \Phi _{1}\Big ( \frac{1}{2}, 1; 1; \frac{-4\,\sqrt{\lambda x}}{(\sqrt{\lambda }-\sqrt{x}\,)^{2}}, -2 \sqrt{\lambda x}\Big )\Big ], &{}\quad x >\lambda \end{array}\right. } . \end{aligned}$$
(11)

Proof

Firstly, we prove that for all \(\lambda , p, x>0\) there holds the relation

$$\begin{aligned} {\mathscr {I}}_p := \int _0^1 \dfrac{\,\mathrm{e}^{-\frac{p}{2} \sqrt{\lambda x}\,t} \mathrm{d}t}{\sqrt{t-t^2}\,(1+ q_p \,t)} = \pi \,\Phi _1\Big ( \frac{1}{2}, 1; 1; -q_p, -\frac{p}{2} \sqrt{\lambda x}\Big ), \end{aligned}$$
(12)

where the shorthand

$$\begin{aligned} q_{p} := \frac{p\,\sqrt{\lambda x}}{(\sqrt{\lambda }-\sqrt{x}\,)^{2}} < 1. \end{aligned}$$

Indeed, the integrand f(t) in (12) has the asymptotic

$$\begin{aligned} f(t) \sim {\left\{ \begin{array}{ll} t^{-\frac{1}{2}}, &{}\quad t \rightarrow 0+ \\ \dfrac{(1-t)^{-\frac{1}{2}}\,\mathrm{e}^{-\frac{p}{2} \sqrt{\lambda x}}}{1+q_p}, &{}\quad t \rightarrow 1- \end{array}\right. } , \end{aligned}$$

which provides the convergence of \({\mathscr {I}}_p\) being an Euler-type single integral representation of the Humbert \(\Phi _1\) function [5, p. 348, Eq. (9)]

$$\begin{aligned} \Phi _1\big ( \alpha , \beta ; \gamma ; z, w) = \dfrac{\Gamma (\gamma )}{\Gamma (\alpha )\,\Gamma (\gamma -\alpha )} \int _0^1 t^{\alpha -1} (1-t)^{\gamma -\alpha -1} (1-z t)^{-\beta }\,\mathrm{e}^{w\,t}\, \mathrm{d}t, \end{aligned}$$

for all \(\Re (\gamma )> \Re (\alpha ) >0\). This integral form is obtained from the on-dimensional Picard–Goursat Euler-type integral of the Appell double hypergeometric function \(F_1\) by confluence, which is

$$\begin{aligned} \Phi _1\big ( \alpha , \beta ; \gamma ; z, w) = \lim _{\epsilon \rightarrow 0} F_1 \big ( \alpha , \beta , \varepsilon ^{-1}; \gamma ; x, \varepsilon y\big ).\end{aligned}$$

Now, equalizing the parameters’ values of the integrands in \(\Phi _1\) and in

$$\begin{aligned} {\mathscr {I}}_p = \dfrac{1}{(\sqrt{\lambda }- \sqrt{x}\,)^2} \int _0^1 t^{-\frac{1}{2}} (1-t)^{-\frac{1}{2}} (1+q_p t)^{-1} \mathrm{e}^{-\frac{p}{2} \sqrt{\lambda x}\,t} \,\mathrm{d}t,\end{aligned}$$

we immediately get the parameters values

$$\begin{aligned} \alpha = \frac{1}{2},\, \beta = 1,\, \gamma = 1;\, z = -q_p;\, w = -\frac{p\, \sqrt{\lambda x}}{2}.\end{aligned}$$

Hence, we have

$$\begin{aligned} {\mathscr {I}}_p&= \dfrac{\pi }{(\sqrt{\lambda }- \sqrt{x}\,)^2}\, \dfrac{\Gamma (1)}{\Gamma ^2(\frac{1}{2})} \int _0^1 t^{-\frac{1}{2}} (1-t)^{-\frac{1}{2}} (1+q_p t)^{-1} \mathrm{e}^{-\frac{p}{2} \sqrt{\lambda x}\,t} \,\mathrm{d}t \\&= \frac{\pi }{(\sqrt{\lambda }- \sqrt{x}\,)^2}\, \Phi _1\Big ( \frac{1}{2}, 1; 1; - q_p, -\frac{p}{2} \sqrt{\lambda x}\Big ). \end{aligned}$$

To recover Goldstein’s formula (8), in our setting (9), it is sufficient to specify \(p=4\) in (12). So, for all \(\lambda , x >0\), such that \(q_4<1\), there holds the relation

$$\begin{aligned} {\mathscr {I}}_{4} = \dfrac{\pi }{(\sqrt{\lambda }-\sqrt{x}\,)^{2}}\, \Phi _{1}\Big ( \frac{1}{2}, 1; 1;- \frac{4\,\sqrt{\lambda x}}{(\sqrt{\lambda }-\sqrt{x}\,)^{2}},\, -2\sqrt{\lambda x}\Big ). \end{aligned}$$

Having in mind all the relations (6)–(9), then (12), we start with the integral \({\mathscr {I}}_4\), which precises Goldstein’s auxiliary function \(I(\xi , \eta )\) in (8) via \(\Phi _1\). This in turn leads to the established form (11). It just remains to replace that \(Q_1\) into (10) to complete the proof of the theorem. \(\square \)

Remark 1

The formula (12) can be reached by expanding \(\exp \big (-\frac{p}{2} \sqrt{\lambda x}\,t\big )\) into Maclaurin series and apply the readily justified interchange of summation and integration and the linear argument transformation of \({}_2F_1\). Therefore,

$$\begin{aligned} {\mathscr {I}}_p&= \dfrac{\pi }{(\sqrt{\lambda }-\sqrt{x}\,)^2} \sum _{n \ge 0}\dfrac{\left( \frac{1}{2}\right) _n\, \big (-\frac{p}{2} \sqrt{\lambda x}\,\big )^n}{(1)_n\,n!}\, {}_2F_1\Big ( \begin{array}{c} 1,n+\frac{1}{2}\\ n+1 \end{array} \Big |-q_p\Big ) \\ \nonumber&= \dfrac{\pi }{(\sqrt{\lambda }-\sqrt{x}\,)^2} \sum _{k, n \ge 0}\dfrac{\left( \frac{1}{2}\right) _{k+n} (1)_k}{(1)_{k+n}}\, \dfrac{(-q_p)^k}{k!} \dfrac{\big (-\frac{p}{2} \sqrt{\lambda x}\,\big )^n}{n!}, \end{aligned}$$

where the formula \((a)_n (a+n)_k = (a)_{n+k}\) was used. Hence,

$$\begin{aligned} {\mathscr {I}}_p = \dfrac{\pi }{(\sqrt{\lambda }-\sqrt{x})^2}\, \Phi _1\Big ( \frac{1}{2}, 1; 1; - \frac{p\,\sqrt{\lambda x}}{(\sqrt{\lambda }-\sqrt{x}\,)^2}, -\frac{p}{2} \sqrt{\lambda x}\Big ); \end{aligned}$$

the Humbert series converges since \(q_p<1\).

Remark 2

The new representation of Goldstein’s function \(I(\xi ,\eta )\) with \(\xi ^{\prime }= \sqrt{\lambda x}\), \(\eta ^{\prime }=\sqrt{x/\lambda }\) we get for all \(4 \eta ^{\prime }(1-\eta ^\prime )^{-2}<1\) and \(\xi ^{\prime }>0\), that

$$\begin{aligned} I(\xi ^{\prime },\eta ^{\prime }) = \frac{1}{2} \Big [ I_{0}(\xi ^{\prime })+\frac{(1+\eta ^{\prime })\, \mathrm{e}^{\xi ^{\prime }}}{1-\eta ^{\prime }}\,\Phi _{1}\Big (\frac{1}{2}, 1; 1;-\frac{4\,\eta ^{\prime }}{(1-\eta ^\prime )^2},-2\, \xi ^{\prime }\Big )\Big ].\end{aligned}$$

Remark 3

The cases in (11) are connected via the property [13, p. 2158, Eq. (25)]

$$\begin{aligned} Q_{1}(a,b)+Q_{1}(b,a) = 1+\mathrm{e}^{-\frac{a^{2}+b^{2}}{2}}\,I_{0}(ab),\end{aligned}$$

since the right-hand side expressions are invariant with respect to \(\lambda \leftrightarrow x\) exchange.

The region of validity of Theorem 1 is restricted by \(q_{4}<1\) for \(\lambda , x>0\) determined in the convergence domain of \(\Phi _{1}\). Our aim is to extend the parameter space to negative values of \(\lambda \) and x and inter-connect \(Q_{1}\) and \(\Phi _{1}\) in two different ways. However, the representation formula which expresses Humbert \(\Phi _{1}\) function in terms of \(Q_{1}\), holds for all complex values of variables with the standard restriction for the first argument of \(\Phi _{1}\).

In order to derive our next result, we need the definition of the incomplete Lipschitz–Hankel integrals (ILHI) [16, p. 2815, Eq. (1)]

$$\begin{aligned} I_{e_{\mu ,\nu }}(\alpha ,z) = \int _{0}^z t^{\mu }\,\mathrm{e}^{-\alpha t}\, I_{\nu }(t)\,\mathrm{d}t, \end{aligned}$$

where the integral \(I_{e_{\mu ,\nu }}(\alpha ,z)\) converges provided \(\Re (\mu +\nu )>-1\). Also, we are interested in Rice \(I_{e}\)-function, defined for all \(x \ge 0\) and \(0 \le k\le 1\) by the following finite integral representation [22, p. 7805, Eq. (57)]

$$\begin{aligned} I_{e}(k,x) = \int _0^x \mathrm{e}^{-t}I_{0}(kt)\, \mathrm{d}t. \end{aligned}$$

Theorem 2

For all \(x, \lambda \in {\mathbb {R}}\) for which \(4|\lambda x|(\lambda -x)^{-2}<1\), we have

$$\begin{aligned} Q_{1}(x, \lambda ) = \frac{1}{2}\,\mathrm{e}^{-\frac{x^{2} + \lambda ^{2}}{2}} \,\Big [ I_{0}(\lambda x) + \dfrac{\lambda +x}{\lambda -x}\, \mathrm{e}^{\lambda x}\, \Phi _{1}\Big (\frac{1}{2}, 1; 1; \dfrac{-4 \lambda x}{(\lambda -x)^{2}}, -2 \lambda x\Big )\Big ].\nonumber \\ \end{aligned}$$
(13)

Moreover, for all \(w, z \in {\mathbb {C}}\) such that \(|w|<1, |\Re (z)|<\infty \), there holds true

$$\begin{aligned} \Phi _{1}\Big ( \frac{1}{2}, 1; 1; w, z\Big )&= - \dfrac{\mathrm{e}^{\frac{z}{2}}I_{0}\left( \frac{z}{2}\right) }{\sqrt{1-w}}\, + \dfrac{2\, \mathrm{e}^{\frac{z}{w}}}{\sqrt{1-w}}\nonumber \\&\quad \times Q_{1}\Big ( \big ( 1-\sqrt{1-w}\,\big ) \sqrt{\frac{z}{2 w}},\,\big ( 1+\sqrt{1-w}\,\big ) \sqrt{\frac{z}{2 w}}\,\Big ). \end{aligned}$$
(14)

Proof

The basic connection formula between the zero order ILHI \(I_{e_{0,0}}\), and the Rice \(I_e\)-function reads [16, p. 2822]

$$\begin{aligned} I_{e_{0,0}}(\alpha , x) = \alpha ^{-1} I_e(\alpha ^{-1}, \alpha x). \end{aligned}$$

Combining the representation of \(I_e\) in terms of Marcum \(Q_1\) obtained by Paris et al. [16, p. 2816, Lemma 3] and the ILHI via Humbert \(\Phi _1\) series established in Sofotasios et al. [22, p. 7806, Theorem 6] we readily conclude

$$\begin{aligned} Q_1\Big ( \sqrt{\dfrac{t}{\widetilde{\alpha }}}, \sqrt{t \widetilde{\alpha }}\,\Big ) = \frac{1}{2}\,\mathrm{e}^{-\alpha t}\, \Big [ I_0(t) + \sqrt{\frac{\alpha -1}{\alpha +1}}\, \mathrm{e}^{-t}\, \Phi _1\Big ( \frac{1}{2}, 1; 1; \frac{2}{\alpha +1}, 2 t\Big )\Big ], \end{aligned}$$
(15)

valid for all \(\widetilde{\alpha } = \alpha +\sqrt{\alpha ^2-1}, \, \alpha \ge 1\) and \(t>0\). However, by scaling \(t = -\lambda x\) and \(\alpha = - \frac{x^2+\lambda ^2}{2 \lambda x}\), we arrive at the first statement (13). By virtue of the substitutions \(w = 2(1+\alpha )^{-1}, z = 2t\) in (15) we conclude the desired extension formula (14) after some routine calculations. \(\square \)

Remark 4

We point out that Lemma 3 (i) of [16, p. 2816] is erroneous. Namely, the denominator of the first variable of Marcum \(Q_{1}\) should be correctly written as \(\sqrt{\alpha + \sqrt{\alpha ^{2}-1}}\) according to other three (ii)–(iv) statements of the Lemma 3. Also, the correct value of the parameter \( {\overline{\alpha }} = 1/\sqrt{\alpha ^{2}-1}\) throughout.

Finally, recalling the formula (14) and making use of the representation (4) of the Marcum \(Q_{1}\) function through \(\Phi _{3}\), one can readily find the connection formulae between \(\Phi _{1}\) and \(\Phi _{3}\).

Theorem 3

For all \(z, w \in {\mathbb {C}}; |w|<1\) we have

$$\begin{aligned} \Phi _{1}\Big (\frac{1}{2}, 1; 1; w, z \Big ) = - \frac{\mathrm{e}^{\frac{z}{2}}}{\sqrt{1-w}} \Big [\,I_{0}\Big (\frac{z}{2} \Big ) + 2\,\Phi _{3}\Big (1; 1; \dfrac{wz}{4(1+\sqrt{1-w})^{2}}, \frac{z^{2}}{16}\Big )\Big ]. \end{aligned}$$

Moreover, there holds true the vice versa formula

$$\begin{aligned} \Phi _{3}\Big ( 1; 1; w, z\Big )&= \frac{1}{2} I_{0}\big ( 2\sqrt{z}\,\big ) \\&\quad + \frac{1}{2} \sqrt{\frac{(w+\sqrt{z})^{2}}{(w-\sqrt{z})^{2}}}\, \mathrm{e}^{2\sqrt{z}}\,\Phi _{1}\Big ( \frac{1}{2}, 1; 1; \frac{-4w\sqrt{z}}{(w-\sqrt{z}\,)^{2}},-4 \sqrt{z} \Big ), \end{aligned}$$

valid for \(w, z \in {\mathbb {C}}\), provided \(4|w\sqrt{z}|\,|w-\sqrt{z}\,|^{-2} <1\). Here the reduction of the square root is not allowed, since the complex radicand.

3 Conclusions: An Open Problem

The CDF of the \(\chi _\nu '^2(\lambda )\) distribution is expressed in terms of specific, equal-parameter Humbert \(\Phi _1\) function, with the aid of the Marcum function of the first order \(Q_1\), using the auxiliary formulae J(u, v) and \(I(\xi , \eta )\) of Goldstein, see Theorem 1 in light of Remark 2. In turn, since \(Q_1\) is expressible via specific Humbert functions \(\Phi _1\) and \(\Phi _3\) (see Theorem 2 and (4)), we deduce transformation formulae between the specific cases of \(\Phi _1\) and \(\Phi _3\) in both directions, which are novelties in the theory of hypergeometric functions of two variables (Theorem 3). Here we point out that the used \(\Phi _3\) in our results has equal (unit) parameters, but their nature differ—one stands for the double-indices Pochhammer symbol \((1)_{k+n}\), while the other \((1)_k\), see (4). The equal, also unit parameters in \(\Phi _1\) are of the same fashion, both build single-indexed Pochhammer symbols starting with the expression (5) concluding with the Theorem 3.

The contributions of this study to the theory of probability distributions and to the related area of higher transcendental hypergeometric functions of two variables is therefore evident.

Moreover, the question of either reducing and/or approximating Marcum functions of higher order by a single variable special functions with arbitrary accuracy, still remains open.