Barnes–Ismagilov Integrals and Hypergeometric Functions of the Complex Field

Barnes–Ismagilov Integrals and Hypergeometric
Functions of the Complex Field††This paper is a contribution to the Special
Issue on Elliptic Integrable Systems, Special Functions and Quantum Field
Theory. The full collection is available at
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html

Yury A. NERETIN ${}^{{\dagger}^{1}{\dagger}^{2}{\dagger}^{3}{\dagger}^{4}}$

Yu.A. Neretin

${}^{{\dagger}^{1}}$ Wolfgang Pauli Institut, c/o Fakultät für Mathematik,
Universität Wien,
${}^{{\dagger}^{1}}$ Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
<EMAIL_ADDRESS>http://mat.univie.ac.at/~neretin/
${}^{{\dagger}^{2}}$ Institute for Theoretical and Experimental Physics,
Moscow, Russia ${}^{{\dagger}^{3}}$ Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University, Russia ${}^{{\dagger}^{4}}$ Institute for
Information Transmission Problems, Moscow, Russia

Received April 09, 2020, in final form July 17, 2020; Published online August
02, 2020

We examine a family
${}_{p}G_{q}^{\mathbb{C}}\big{[}\genfrac{}{}{0.0pt}{}{(a)}{(b)};z\big{]}$ of
integrals of Mellin–Barnes type over the space
${\mathbb{Z}}\times{\mathbb{R}}$, such functions $G$ naturally arise in
representation theory of the Lorentz group. We express
${}_{p}G_{q}^{\mathbb{C}}(z)$ as quadratic expressions in the generalized
hypergeometric functions ${}_{p}F_{q-1}$ and discuss further properties of the
functions ${}_{p}G_{q}^{\mathbb{C}}(z)$.

Mellin–Barnes integrals; Mellin transform; hypergeometric functions; Lorentz
group

33C20; 33C70; 22E43

##  1 The statements

###  1.1 Introduction

Recall the Euler integral representation of the Gauss hypergeometric function:

$\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a,b}{c};z\right]=\frac{1}{\mathrm{B}(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,{\rm
d}t,$ (1.1)

where $a$, $b$, $c$ are complex numbers and

$\mathrm{B}(\alpha,\beta):=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\,{\rm
d}t=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)},$ (1.2)

is the beta function. The hypergeometric functions ${}_{p+1}F_{p}$ admit the
following inductive integral representation:

$\displaystyle\fourIdx{}{p+1}{}{\\!p}F\left[\genfrac{}{}{0.0pt}{}{a_{1},\dots,a_{p},c}{b_{1},\dots,b_{p-1},d};z\right]:=\frac{1}{\mathrm{B}(c,d-c)}\\!\int_{0}^{1}\\!t^{c-1}(1-t)^{d-c-1}\fourIdx{}{p}{}{\\!p-1}F\\!\left[\genfrac{}{}{0.0pt}{}{a_{1},\dots,a_{p}}{b_{1},\dots,b_{p-1}};tz\right]{\rm
d}t.\\!\\!\\!\\!\\!$ (1.3)

Let us replace the integration over the interval $[0,1]$ by the integration
over the complex plane for all integrals (1.1)–(1.3). The expression $t^{b}$
for $t\in{\mathbb{C}}$, $b\in{\mathbb{C}}$ is ramified at $t=0$, so the
integration makes no sense. But we can replace $t^{b}$ by
$|t|^{2b}=t^{b}\overline{t}^{b}$. It is better to consider a wider generality
and to write

$t^{b|b^{\prime}}:=t^{b}\overline{t}^{b^{\prime}}.$

This expression is non-ramified if $b-b^{\prime}\in{\mathbb{Z}}$. Then the new
beta-function equals (see [17, Section II.3.7], [16, Section 1.6])111To
evaluate this integral, we can pass to polar coordinates $t=r{\rm e}^{{\rm
i}\varphi}$. Integrating in $\varphi$ we get a piece-wise
${}_{2}F_{1}$-expression in $r$ with a singularity at $r=1$. It remains to
apply the Slater theorem, see [43, Theorem 4.8.1, namely formula (4.8.7)]. On
the other hand, we can reduce (1.4) to the gamma-function (1.7) in the usual
way (see Jacobi’s proof of the beta-integral (1.2) in [1, Section I.1]), but a
justification of a changing of integration order is unexpectedly tedious.

$\displaystyle\mathrm{B}^{\mathbb{C}}(\alpha|\alpha^{\prime},\beta|\beta^{\prime})$
$\displaystyle:=\frac{1}{\pi}\int_{\mathbb{C}}t^{\alpha-1|\alpha^{\prime}-1}(1-t)^{\beta-1|\beta^{\prime}-1}\,{\rm
d}\mathop{\mathrm{Re}}\nolimits t\,{\rm d}\mathop{\mathrm{Im}}\nolimits t$
$\displaystyle\hphantom{:}=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\cdot\frac{\Gamma(1-\alpha^{\prime}-\beta^{\prime})}{\Gamma(1-\alpha^{\prime})\Gamma(1-\beta^{\prime})}.$
(1.4)

The new version of the Gauss hypergeometric function (Gauss hypergeometric
functions of the complex field) also can be evaluated (see [33, Theorem 3.9]),
the result has the form

$\displaystyle\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a|a^{\prime},b|b^{\prime}}{c|c^{\prime}};z\right]=\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a,b}{c};z\right]\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a^{\prime},b^{\prime}}{c^{\prime}};\overline{z}\right]$
$\displaystyle{}+\left\\{\begin{matrix}\text{product of}\\\
\text{$\Gamma$-functions}\end{matrix}\right\\}z^{1-c|1-c^{\prime}}\fourIdx{}{2}{}{\\!1}F\\!\left[\genfrac{}{}{0.0pt}{}{a+1-c,b+1-c}{2-c};z\right]\fourIdx{}{2}{}{\\!1}F\\!\left[\genfrac{}{}{0.0pt}{}{a^{\prime}+1-c^{\prime},b^{\prime}+1-c^{\prime}}{2-c^{\prime}};\overline{z}\right].\\!\\!\\!$
(1.5)

The Gauss hypergeometric functions satisfy numerous identities, see for
instance the book Higher Hypergeometric Functions [13, Chapter 2]. Usually
such identities (and maybe all such identities) have counterparts for
${}_{2}F_{1}^{\mathbb{C}}$ (see a collection of formulas in [33, Section 3]).

Counterparts of the Gauss hypergeometric functions were briefly mentioned in
the book by Gelfand, Graev and Vilenkin [17, Section II.3.7] in 1962. Later
various formulas and identities related to hypergeometric functions of complex
fields of different levels and beta integrals of the complex field appeared in
works of numerous authors: Dotsenko and Fateev [12], Gelfand, Graev and Retakh
[16], Ismagilov [23, 24], Derkachov and Manashov [6, 8], Bazhanov, Mangazeev
and Sergeev [2], Derkachov, Manashov, Valinevich [9, 10], Kels [27, 28],
Mimachi [32], Derkachov and Spiridonov [11], Molchanov and Neretin [33],
Neretin [39] (we discuss the references in a more arranged form in Section
3.3).

The purpose of this paper is to give a definition of counterparts of higher
hypergeometric functions ${}_{p}F_{q}(z)$ (formula (1.15)) and to obtain an
analog of formula (1.5), see Theorem 1.2. The main statements are contained in
the present section, their proofs are done in Section 2. In Section 3 we
discuss further properties of our functions.

###  1.2 Notation

Denote by ${\mathbb{Z}}_{+}$ the set of integers $\geqslant 0$, by
${\mathbb{Z}}_{-}$ the set of integers $\leqslant 0$.

####  1.2.1 Hypergeometric functions.

Let $r\leqslant s+1$. Let $a_{1},\dots,a_{r}\in{\mathbb{C}}$ and
$b_{1},\dots,b_{s}\in{\mathbb{C}}\setminus{\mathbb{Z}}_{-}$. Generalized
hypergeometric functions are defined by

$\fourIdx{}{r}{}{\\!s}F\left[\genfrac{}{}{0.0pt}{}{a_{1},\dots,a_{r}}{b_{1},\dots,b_{r}};z\right]:=\sum_{m=0}^{\infty}\frac{(a_{1})_{m}\cdots(a_{r})_{m}}{m!(b_{1})_{m}\cdots(b_{s})_{m}}\cdot
z^{n},$ (1.6)

where $(a)_{m}:=a(a+1)\cdots(a+m-1)$ is the Pochhammer symbol. For $r\leqslant
s$ the radius of convergence is $\infty$, for $r=s+1$ it is $1$. For $r>s+1$
the series diverges.

####  1.2.2 Notation for lists

Let $a_{1},\dots,a_{p}\in{\mathbb{C}}$ and $h\in{\mathbb{C}}$. We use the
following notation for lists,

$\displaystyle(a):=a_{1},\dots,a_{p};$
$\displaystyle(a)+h:=a_{1}+h,\dots,a_{p}+h;$ $\displaystyle(a)_{\setminus
j}:=a_{1},\dots,a_{j-1},a_{j+1},\dots,a_{p}.$

In particular, we denote (1.6) by
$\fourIdx{}{r}{}{\\!s}F\big{[}\genfrac{}{}{0.0pt}{}{(a)}{(b)};z\big{]}$.

####  1.2.3 Double powers

Denote by ${\mathbb{C}}^{\times}$ (resp. ${\mathbb{R}}^{\times}_{+}$) the
multiplicative group of ${\mathbb{C}}$ (resp. the multiplicative group of
positive reals). By $\mathbb{T}\subset{\mathbb{C}}^{\times}$ we denote the
subgroup $|z|=1$, we have $\mathbb{T}\simeq{\mathbb{R}}/2\pi{\mathbb{Z}}$,

${\mathbb{C}}^{\times}=\mathbb{T}\times{\mathbb{R}}^{\times}_{+},$

and ${\mathbb{R}}^{\times}_{+}$ is isomorphic to the additive group
${\mathbb{R}}$ of real numbers.

The dual group $({\mathbb{C}}^{\times})^{*}$, i.e., the group of all
homomorphisms ${\mathbb{C}}^{\times}\to\mathbb{T}$, is

$({\mathbb{C}}^{\times})^{*}=(\mathbb{T})^{*}\times({\mathbb{R}}^{\times}_{+})^{*}\simeq{\mathbb{Z}}\times{\mathbb{R}}.$

Denote by $\Lambda_{\mathbb{C}}$ the set of pairs $a\big{|}a^{\prime}$ such
that $a$, $a^{\prime}\in{\mathbb{C}}$ and $a-a^{\prime}\in{\mathbb{Z}}$. We
write such pairs by

$a\big{|}a^{\prime}=\tfrac{k+\sigma}{2}\big{|}\tfrac{-k+\sigma}{2},\qquad\text{where
$k\in{\mathbb{Z}}$, $\sigma\in{\mathbb{C}}$}.$

By $\Lambda\subset\Lambda_{\mathbb{C}}$ we denote the subset consisting of
$a|a^{\prime}$ satisfying $a^{\prime}+\overline{a}=0$,

$a\big{|}a^{\prime}=\tfrac{k+{\rm i}s}{2}\big{|}\tfrac{-k+{\rm
i}s}{2}\in\Lambda,\qquad\text{where $s\in{\mathbb{R}}$}.$

Define the following functions on ${\mathbb{C}}^{\times}$:

$z^{a|a^{\prime}}:=z^{a}\overline{z}^{\,a^{\prime}}:=(z/\overline{z})^{k/2}|z|^{\sigma}.$

Notice that such functions are precisely all homomorphisms
${\mathbb{C}}^{\times}\to{\mathbb{C}}^{\times}$. For $a|a^{\prime}\in\Lambda$
we get homomorphisms ${\mathbb{C}}^{\times}\to\mathbb{T}$,

$\Big{|}z^{\tfrac{k+{\rm i}s}{2}\big{|}\tfrac{-k+{\rm i}s}{2}}\Big{|}=1.$

####  1.2.4 The complex Mellin transform

Denote by $\,{\rm d}\,{\overline{\overline{t}}}$ the Lebesgue measure on
${\mathbb{C}}$,

$\,{\rm d}\,{\overline{\overline{t}}}:={\rm d}\mathop{\mathrm{Re}}\nolimits
t\,{\rm d}\mathop{\mathrm{Im}}\nolimits t.$

We define the complex Mellin transform $\mathcal{M}$ as the Fourier transform
on the group ${\mathbb{C}}^{\times}$. For a function $f$ on
${\mathbb{C}}^{\times}$ we define the Mellin transform as a function on
$\Lambda_{\mathbb{C}}$ defined by

$g\big{(}a\big{|}a^{\prime}\big{)}=\mathcal{M}f\big{(}a\big{|}a^{\prime}\big{)}:=\frac{1}{2\pi}\int_{{\mathbb{C}}^{\times}}t^{a|a^{\prime}}f(t)\frac{\,{\rm
d}\,{\overline{\overline{t}}}}{t^{1|1}},$

the factor ${\,{\rm d}\,{\overline{\overline{t}}}}/{t^{1|1}}$ is the
${\mathbb{C}}^{\times}$-invariant measure on ${\mathbb{C}}^{\times}$.

The Mellin transform is a unitary operator from
$L^{2}\big{(}{\mathbb{C}}^{\times},{\,{\rm
d}\,{\overline{\overline{t}}}}/{t^{1|1}}\big{)}$ to $L^{2}$ on
$\Lambda\simeq{\mathbb{Z}}\times{\mathbb{R}}$, the inversion formula is

$f(t)=\frac{1}{2\pi}\sum_{k\in{\mathbb{Z}}}\int_{\mathbb{R}}g\big{(}\tfrac{k+{\rm
i}s}{2}\big{|}\tfrac{-k+{\rm i}s}{2}\big{)}t^{{\tfrac{-k-{\rm
i}s}{2}\big{|}\tfrac{k-{\rm i}s}{2}}}\,{\rm d}s.$

The convolution on the group ${\mathbb{C}}^{\times}$ is defined by formula

$f*g(z):=\int_{\mathbb{C}}f(t)g(z/t)\,\frac{\,{\rm
d}\,{\overline{\overline{t}}}}{t^{1|1}}.$

The Mellin transform sends convolutions to products,

$\mathcal{M}(f*g)(z)=2\pi\,\mathcal{M}f(z)\cdot\mathcal{M}g(z).$

###  1.3 The Gamma-function of the complex field

Following Gelfand, Graev and Retakh [16], we define the Gamma-function of the
complex field as a function on $\Lambda_{\mathbb{C}}$ by222The definition
slightly differs from a definition from [16], which was used in [33, 39]. We
write $\mathop{\mathrm{Im}}\nolimits t$ instead of
$\mathop{\mathrm{Re}}\nolimits t$. For this reason the factor
$i^{a-a^{\prime}}$ from [16, 33] disappears.

$\displaystyle\Gamma^{\mathbb{C}}\big{(}a\big{|}a^{\prime}\big{)}$
$\displaystyle:=\frac{1}{\pi}\int_{\mathbb{C}}t^{a-1|a^{\prime}-1}{\rm
e}^{2{\rm i}\mathop{\mathrm{Im}}\nolimits t}\,{\rm
d}\,{\overline{\overline{t}}}a$
$\displaystyle\hphantom{:}=\frac{1}{\pi}\lim_{r\to\infty}\int_{|t|\leqslant
r}t^{a-1|a^{\prime}-1}{\rm e}^{2{\rm i}\mathop{\mathrm{Im}}\nolimits t}\,{\rm
d}\,{\overline{\overline{t}}}=\frac{\Gamma(a)}{\Gamma(1-a^{\prime})}=\frac{(-1)^{a-a^{\prime}}\Gamma(a^{\prime})}{\Gamma(1-a)}$
$\displaystyle\hphantom{:}=\frac{1}{\pi}\,\Gamma(a)\Gamma(a^{\prime})\sin\pi
a^{\prime}=\frac{(-1)^{a-a^{\prime}}}{\pi}\,\Gamma(a)\Gamma(a^{\prime})\sin\pi
a.$ (1.7)

The integral conditionally converges if
$0<\mathop{\mathrm{Re}}\nolimits(a+a^{\prime})<1$ and diverges otherwise.
Clearly, the right hand side is meromorphic in the whole
$\Lambda_{\mathbb{C}}$.

######  Remark 1.1.

In particular, we can write (1.4) as

$\mathrm{B}^{\mathbb{C}}(\alpha|\alpha^{\prime},\beta|\beta^{\prime})=\frac{\Gamma^{\mathbb{C}}(\alpha|\alpha^{\prime})\,\Gamma^{\mathbb{C}}(\beta|\beta^{\prime})}{\Gamma^{\mathbb{C}}(\alpha+\beta|\alpha^{\prime}+\beta^{\prime})}.$

It is easy to see that

$\displaystyle\Gamma^{\mathbb{C}}\big{(}a|a^{\prime}\big{)}=(-1)^{a-a}\Gamma^{\mathbb{C}}\big{(}a^{\prime}|a\big{)},$
(1.8a)
$\displaystyle\Gamma^{\mathbb{C}}\big{(}a|a^{\prime}\big{)}\,\Gamma^{\mathbb{C}}\big{(}1-a|1-a^{\prime}\big{)}=(-1)^{a-a^{\prime}},$
(1.8b)
$\displaystyle\Gamma^{\mathbb{C}}\big{(}a+m\big{|}a^{\prime}+m^{\prime}\big{)}=(-1)^{m^{\prime}}\Gamma^{\mathbb{C}}\big{(}a\big{|}a^{\prime}\big{)}(a)_{m}(a^{\prime})_{m^{\prime}},$
(1.8c)
$\displaystyle\Gamma^{\mathbb{C}}\big{(}a-m\big{|}a^{\prime}-m^{\prime}\big{)}=\frac{(-1)^{m}\Gamma^{\mathbb{C}}\big{(}a\big{|}a^{\prime}\big{)}}{(1-a)_{m}(1-a^{\prime})_{m^{\prime}}},$
(1.8d)
$\displaystyle\prod_{j=0}^{k-1}\Gamma^{\mathbb{C}}\big{(}a+\tfrac{j}{k}\big{|}a^{\prime}+\tfrac{j}{k}\big{)}=\Gamma^{\mathbb{C}}\big{(}ka\big{|}ka^{\prime}\big{)}k^{1-(a+a^{\prime})k},$
(1.8e)

where $m\in{\mathbb{N}}$, $k\in{\mathbb{N}}$. Values of $\Gamma^{\mathbb{C}}$
at integer points are

$\displaystyle\Gamma^{\mathbb{C}}(k|l)=0,\qquad\text{for $k$,
$l\in{\mathbb{N}}$;}$
$\displaystyle\Gamma(m|-k)=\frac{(m-1)!(-1)^{k}}{k!},\qquad\text{for
$m\in{\mathbb{N}}$, $k\in{\mathbb{Z}}_{+}$}.$

For $m$, $m^{\prime}\in{\mathbb{Z}}_{+}$ we have a pole at
$(-m)|(-m^{\prime})$, more precisely,

$\mathop{\mathrm{res}}\nolimits\limits_{\varepsilon=0}\Gamma^{\mathbb{C}}(-m+\varepsilon|-m^{\prime}+\varepsilon)=\frac{(-1)^{m}}{m!\,m^{\prime}!}.$
(1.9)

Next (see Section 2.1 below), for $a|a^{\prime}\in\Lambda_{\mathbb{C}}$ and
$\xi=\frac{1}{2}(k+{\rm i}s)$ with $k\in{\mathbb{Z}}$, $s\in{\mathbb{R}}$ we
have

$\Gamma^{\mathbb{C}}\big{(}a+\xi|a^{\prime}-\overline{\xi}\big{)}=\exp\big{\\{}2{\rm
i}\mathop{\mathrm{Im}}\nolimits(\xi\ln\xi-\xi)\big{\\}}\cdot\xi^{a-\tfrac{1}{2}\big{|}a^{\prime}-\tfrac{1}{2}}\big{(}1+O\big{(}|\xi|^{-1}\big{)}\big{)}.$
(1.10)

Therefore,

$\big{|}\Gamma^{\mathbb{C}}\big{(}a+\xi|a^{\prime}-\overline{\xi}\big{)}\big{|}\sim|\xi|^{\mathop{\mathrm{Re}}\nolimits(a+a^{\prime})-1}.$
(1.11)

###  1.4 Hypergeometric functions of the complex field

Let $a_{1}|a^{\prime}_{1},\dots,a_{q}|a^{\prime}_{q}$,
$b_{1}|b^{\prime}_{1},\dots,b_{p}|b^{\prime}_{p}\in\Lambda_{\mathbb{C}}$.
Temporarily, we assume that they satisfy the conditions:

$\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})>0,\qquad\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})>0.$
(1.12)

and

$\sum_{\alpha}\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})+\sum_{\beta}\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})<p+q.$
(1.13)

We define the following function on $\Lambda_{\mathbb{C}}$:

$\displaystyle\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};\tfrac{k+\sigma}{2}\Bigl{|}\tfrac{-k+\sigma}{2}\right]$
$\displaystyle\qquad{}:=\prod_{\alpha=1}^{q}\Gamma^{\mathbb{C}}\Bigl{(}a_{\alpha}+\tfrac{k+\sigma}{2}\Bigl{|}a_{\alpha}^{\prime}+\tfrac{-k+\sigma}{2}\Bigr{)}\prod_{\beta=1}^{p}\Gamma^{\mathbb{C}}\Bigl{(}b_{\beta}+\tfrac{-k-\sigma}{2}\Bigl{|}b_{\beta}^{\prime}+\tfrac{k-\sigma}{2}\Bigr{)}.$
(1.14)

Next, we define the hypergeometric functions of the complex field as the
contour integral:

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]:=\frac{1}{2\pi{\rm
i}}\sum_{k\in{\mathbb{Z}}}\int_{{\rm
i}{\mathbb{R}}}\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};\tfrac{k+\sigma}{2}\Bigl{|}\tfrac{-k+\sigma}{2}\right]z^{\frac{-k-\sigma}{2}\big{|}\frac{k-\sigma}{2}}\,{\rm
d}\sigma.$ (1.15)

The integration is taken along the imaginary axis ${\rm i}{\mathbb{R}}$. The
condition (1.13) provides the conditional convergence of the integral in
$\sigma$ and the absolute convergence of the series (see Sections 2.2 and 2.3
below). Under the stronger condition

$\sum_{\alpha}\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})+\sum_{\beta}\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})<p+q-1$
(1.16)

all integrals in $\sigma$ convergence absolutely (this follows from (1.11)).

The expression (1.14) has poles (they are analyzed in Section 2.3) in $\sigma$
originating from two groups of factors. We call poles of the factors
$\Gamma^{\mathbb{C}}\big{(}a+\frac{k+\sigma}{2}\big{|}a^{\prime}+\frac{-k+\sigma}{2}\big{)}$
_left poles_ , the poles of the factors
$\Gamma^{\mathbb{C}}\big{(}b+\frac{-k-\sigma}{2}\big{|}b^{\prime}+\frac{k-\sigma}{2}\big{)}$
_right poles_. Under the conditions (1.12) left poles are contained in the
left half-plane $\mathop{\mathrm{Re}}\nolimits\sigma<0$, right poles are
contained in the right half plane $\mathop{\mathrm{Re}}\nolimits\sigma>0$, and
the axis ${\rm i}{\mathbb{R}}$ separates these groups of poles.

As usual (see [43]) we can write the analytic continuation of
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
to a wider domain of $(a|a^{\prime})$, $(b|b^{\prime})$ by moving integration
contours. Let us omit the condition (1.12) and fix
$a_{\alpha}|a^{\prime}_{\alpha}$, $b_{\beta}|b_{\beta}^{\prime}$. Then ${\rm
i}{\mathbb{R}}$ separates left and right poles for all but a finite number of
summands. Assume

$a_{\alpha}+b_{\beta}\notin{\mathbb{Z}}_{+}\qquad\text{for all $\alpha$,
$\beta$.}$ (1.17)

For each $k$ we choose a contour $L_{k}$ that coincides with ${\rm
i}{\mathbb{R}}$ at infinity and separates left and right poles of the
corresponding summand, the result of the integration does not depend on a
choice of $L_{k}$. Then we replace the expression (1.15) by

$\sum_{k}\int_{L_{k}}$

and get an expression of our integral in the domain defined by conditions
(1.13), (1.17).

###  1.5 The statements of the paper

For the same $(a|a^{\prime})$, $(b|b^{\prime})$ we define the expression
$\Sigma^{\mathbb{C}}_{+}(z)$ by

$\displaystyle\Sigma^{\mathbb{C}}_{+}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]:=2\sum_{j=1}^{q}z^{a_{j}|a_{j}^{\prime}}\cdot\prod_{\beta}\Gamma^{\mathbb{C}}\big{(}b_{\beta}+a_{j}\big{|}b_{\beta}^{\prime}+a_{j}^{\prime}\big{)}\cdot\prod_{\alpha\neq
j}\Gamma^{\mathbb{C}}\big{(}a_{\alpha}-a_{j}\big{|}a_{\alpha}^{\prime}-a_{j}^{\prime}\big{)}$
$\displaystyle\qquad{}\times\fourIdx{}{p}{}{\\!q-1}F\left[\genfrac{}{}{0.0pt}{}{(b_{\alpha}+a_{j})}{(1-a_{\alpha}+a_{j})_{\setminus
j}};(-1)^{q}\,z\right]\fourIdx{}{p}{}{\\!q-1}F\left[\genfrac{}{}{0.0pt}{}{(b_{\alpha}^{\prime}+a_{j}^{\prime})}{(1-a_{\alpha}^{\prime}+a_{j}^{\prime})_{\setminus
j}};(-1)^{p}\,\overline{z}\right].$ (1.18)

We also define the expression $\Sigma^{\mathbb{C}}_{-}(z)$ by

$\Sigma^{\mathbb{C}}_{-}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]:=\Sigma^{\mathbb{C}}_{+}\left[\genfrac{}{}{0.0pt}{}{(b|b^{\prime})}{(a|a^{\prime})};z^{-1}\right].$

######  Theorem 1.2.

Let $(a|a^{\prime})$, $(b|b^{\prime})$ satisfy the conditions (1.12) and
(1.13).

  1. $(a)$

For $q>p$ we have

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\Sigma^{\mathbb{C}}_{+}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]$

and for $q<p$ we have
${}_{p}^{\vphantom{{\mathbb{C}}}}G^{\mathbb{C}}_{q}(z)=\Sigma^{\mathbb{C}}_{-}(z)$.

  2. $(b)$

For $p=q$

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G(z)=\begin{cases}\Sigma^{\mathbb{C}}_{+}(z),&\text{if
$|z|<1$},\\\ \Sigma^{\mathbb{C}}_{-}(z),&\text{if $|z|>1$}.\end{cases}$

######  Remark 1.3.

This statement and the main argument for its proof are potentially contained
in the paper by Ismagilov [24, Lemma 2].

######  Corollary 1.4.

The function
$F(z):=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
satisfies the following system of differential equations

$\mathcal{D}F=0,\qquad\overline{\mathcal{D}}F=0,$ (1.19)

where

$\displaystyle\mathcal{D}:=(-1)^{q}\prod_{\alpha=1}^{p}\left(z\frac{\partial}{\partial
z}+a_{\alpha}\right)-z\prod_{\beta=1}^{q}\left(z\frac{\partial}{\partial
z}-b_{\beta}\right),$ (1.20a)
$\displaystyle\overline{\mathcal{D}}:=(-1)^{p}\prod_{\alpha=1}^{p}\left(\overline{z}\frac{\partial}{\partial\overline{z}}+a_{\alpha}^{\prime}\right)-\overline{z}\prod_{\beta=1}^{q}\left(\overline{z}\frac{\partial}{\partial\overline{z}}-b_{\beta}^{\prime}\right).$
(1.20b)

Consider the expression
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
as a function of a variable $z$ and parameters $(a|a^{\prime})$,
$(b|b^{\prime})$, i.e., a function defined on a certain domain in the space

${\mathbb{C}}\times(\Lambda_{\mathbb{C}})^{p+q}\simeq{\mathbb{C}}\times{\mathbb{Z}}^{p+q}\times{\mathbb{C}}^{p+q}.$

Fixing integer parameters333The conditions (1.12) and (1.13) allow arbitrary
integer parameters. $(a-a^{\prime})$, $(b-b^{\prime})$ we get a countable
collection of functions on domains in ${\mathbb{C}}\times{\mathbb{C}}^{p+q}$.

Next, fix $\alpha$, $\beta$, fix also $m$, $m^{\prime}\in{\mathbb{Z}}_{+}$.
Consider the subset $S(\alpha,\beta;m,m^{\prime})$ in
${\mathbb{C}}\times(\Lambda_{\mathbb{C}})^{p+q}$ defined by the equations

$\begin{cases}a_{\alpha}+b_{\beta}=m,\\\
a_{\alpha}^{\prime}+b_{\beta}^{\prime}=m^{\prime}.\end{cases}$ (1.21)

######  Proposition 1.5.

For fixed $(a-a^{\prime})\in{\mathbb{Z}}^{p+q}$,
$(b-b^{\prime})\in{\mathbb{Z}}^{p+q}$ the expression
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
as a function of $(a)$, $(b)$, $z$ admits an extension to a function which is
real analytic in $z$ (i.e., real analytic as a function in two variables
$\mathop{\mathrm{Re}}\nolimits z$, $\mathop{\mathrm{Im}}\nolimits z$) in the
domain

$\displaystyle z\in{\mathbb{C}}\setminus\\{0\\},\qquad\text{if $p\neq q$},$
$\displaystyle z\in{\mathbb{C}}\setminus\\{0,(-1)^{p}\\},\qquad\text{if
$p=q$},$

and meromorphic in $(a)$, $(b)$ with singularities (simple poles) located in

$\bigcup_{\alpha,\beta,m,m^{\prime}}S(\alpha,\beta;m,m^{\prime}).$

######  Lemma 1.6.

  1. $(a)$

Let the parameters $(a|a^{\prime})$, $(b|b^{\prime})$ satisfy the conditions

$\displaystyle\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})>0,\qquad\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})>0,$
$\displaystyle\sum_{\alpha}\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})+\sum_{\beta}(b_{\beta}+b_{\beta}^{\prime})<p+q-1.$

Then the function
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
is contained in $L^{2}\big{(}{\mathbb{C}},|z|^{-2}\,\,{\rm
d}\,{\overline{\overline{z}}}\big{)}$.

  2. $(b)$

Under the same conditions

$\frac{1}{2\pi}\int_{\mathbb{C}}z^{\sigma-1|\sigma^{\prime}-1}\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]\,\,{\rm
d}\,{\overline{\overline{z}}}=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};\sigma|\sigma^{\prime}\right],$

where the integral is understood as a Mellin transform in $L^{2}$.

######  Theorem 1.7.

Let two functions

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right],\qquad\fourIdx{}{r}{{\mathbb{C}}}{\\!s}G\left[\genfrac{}{}{0.0pt}{}{(c|c^{\prime})}{(d|d^{\prime})};z\right]$

be contained in $L^{2}\big{(}{\mathbb{C}},|z|^{-2}\,{\rm
d}\,{\overline{\overline{z}}}\big{)}$. Then

$\frac{1}{2\pi}\int_{\mathbb{C}}\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]\cdot\fourIdx{}{r}{{\mathbb{C}}}{\\!s}G\left[\genfrac{}{}{0.0pt}{}{(c|c^{\prime})}{(d|d^{\prime})};\frac{t}{z}\right]\frac{\,{\rm
d}\,{\overline{\overline{z}}}}{z^{1|1}}=\fourIdx{}{p+r}{{\mathbb{C}}}{\\!q+s}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime}),(c|c^{\prime})}{(b|b^{\prime}),(d|d^{\prime})};t\right].$
(1.22)

Taking $p=q=1$ (see (3.2)) we get a counterpart of formula (1.3):

$\displaystyle\frac{1}{2\pi}\int_{\mathbb{C}}z^{a|a^{\prime}}(1+z)^{-a-b|-a^{\prime}-b^{\prime}}\cdot\fourIdx{}{r}{{\mathbb{C}}}{\\!s}G\left[\genfrac{}{}{0.0pt}{}{(c|c^{\prime})}{(d|d^{\prime})};\frac{t}{z}\right]\frac{\,{\rm
d}\,{\overline{\overline{z}}}}{z^{1|1}}$
$\displaystyle\qquad{}=\frac{1}{2\Gamma^{\mathbb{C}}(a+b|a^{\prime}+b^{\prime})}\,\fourIdx{}{1+r}{{\mathbb{C}}}{\\!1+s}G\left[\genfrac{}{}{0.0pt}{}{a|a^{\prime},(c|c^{\prime})}{b|b^{\prime},(d|d^{\prime})};t\right].$

##  2 Proofs

###  2.1 Asymptotics of $\boldsymbol{\Gamma^{\mathbb{C}}}$

Let us derive formula (1.10) for the asymptotics of the function
$\Gamma^{C}(a+\xi|a^{\prime}-\overline{\xi})$. First, let us verify that the
right-hand side is single-valued on $\Lambda$. We must verify that the
expression

$\exp\big{\\{}\xi\ln\xi-\overline{\xi}\ln\overline{\xi}\big{\\}},\qquad\xi=\tfrac{k+{\rm
i}s}{2}$

is single valued. Represent $\xi=r{\rm e}^{{\rm i}\varphi}$. We transform our
expression as

$\displaystyle\exp\big{\\{}\xi(\ln(r)+{\rm i}\varphi+2\pi{\rm
i}n)-\overline{\xi}(\ln(r)-{\rm i}\varphi-2\pi{\rm i}n)\big{\\}}$
$\displaystyle\qquad{}=\exp\big{\\{}(\xi-\overline{\xi})\ln
r+(\xi+\overline{\xi})({\rm i}\varphi+2\pi{\rm i}n)\big{\\}}.$

We have $\xi+\overline{\xi}\in{\mathbb{Z}}$ and therefore the exponential is
single valued.

Next, we apply the Stirling formula in the form (see [13, equation (1.18(3))])

$\ln\Gamma(c+z)=\big{(}z+c-\tfrac{1}{2}\big{)}\ln
z-z+\tfrac{1}{2}\ln(2\pi)+O\big{(}z^{-1}\big{)},\qquad\text{where
$|\arg(z)|<\pi-\varepsilon$},$

and get

$\displaystyle\Gamma(a+\xi)\simeq\sqrt{2\pi}\exp\big{\\{}\big{(}\xi+a-\tfrac{1}{2}\big{)}\ln\xi-\xi\big{\\}},$
$\displaystyle\Gamma(1-a^{\prime}-\overline{\xi})\simeq\sqrt{2\pi}\exp\big{\\{}\big{(}\overline{\xi}-a^{\prime}+\tfrac{1}{2}\big{)}\ln\overline{\xi}-\overline{\xi}\big{\\}}.$

The ratio gives us (1.10). An evaluation of the asymptotics in the sector
$|\arg(-z)|<\pi-\varepsilon$ gives the same result.

######  Corollary 2.1.

Let $\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})>0$,
$\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})>0$. Denote

$\upsilon:=\sum(a_{\alpha}+a_{\alpha}^{\prime})+\sum(b_{\beta}+b_{\beta}^{\prime}).$

  1. $(a)$

If $\upsilon<p+q$, then $F(k,{\rm
i}s):=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};\tfrac{k+{\rm
i}s}{2}\bigl{|};\tfrac{-k+{\rm i}s}{2}\big{]}$ tends to $0$ as $|k+{\rm i}s|$
tends to $\infty$.

  2. $(b)$

If $\upsilon<p+q-1$, then for each $k$ the function $F(k,{\rm i}s)$ as a
function in $s$ is integrable. Under the same condition $F(k,{\rm i}s)$ is
contained in $L^{2}(\Lambda)$.

  3. $(c)$

If $\upsilon<p+q-2$, then $F(k,{\rm i}s)$ is contained in $L^{1}(\Lambda)$.

######  Remark 2.2.

Formula (1.10) also gives us asymptotics of
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}$ on vertical lines
$\sigma=h+{\rm i}s$. Indeed,

$\Gamma\big{(}a+\tfrac{k+h+{\rm i}s}{2}\big{|}a^{\prime}+\tfrac{-k+h+{\rm
i}s}{2}\big{)}=\Gamma\big{(}a+\tfrac{h}{2}+\tfrac{k+{\rm
i}s}{2}\big{|}a^{\prime}+\tfrac{h}{2}+\tfrac{-k+{\rm i}s}{2}\big{)},$

and we can control integrability under shifts of the integration contour.

###  2.2 Decomposition of Mellin–Barnes integrals in residues

Now we start to prove Theorem 1.2, i.e., to derive the quadratic expressions
of the functions $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ in terms of the usual
hypergeometric functions $\fourIdx{}{p}{}{\\!q}F$.

First, we write the definition (1.15) of $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$
in the form

$\sum_{k\in{\mathbb{Z}}}z^{-k/2\big{|}k/2}I_{k}(z),\qquad
I_{k}(z)=\frac{1}{2\pi{\rm i}}\int(\dots)|z|^{-\sigma}\,{\rm d}\sigma.$

The integrals $I_{k}(z)$ are special cases of Mellin–Barnes integrals, i.e.,
integrals of the type

$J^{A,B}_{C,D}(a,b,c,d;u)=\frac{1}{2\pi{\rm
i}}\int_{L}\frac{\prod\limits_{\alpha=1}^{A}\Gamma(a_{\alpha}+\sigma)\prod\limits_{\beta=1}^{B}\Gamma(b_{\beta}-\sigma)}{\prod\limits_{\gamma=1}^{C}\Gamma(c_{\gamma}+\sigma)\prod\limits_{\delta=1}^{D}\Gamma(d_{\delta}-\sigma)}u^{-s}\,{\rm
d}s,$ (2.1)

where $L$ is a contour separating poles of factors $\Gamma(a_{\alpha}+\sigma)$
and poles of $\Gamma(b_{\beta}-\sigma)$. The behavior of such Mellin–Barnes
integrals (Meijer $G$-functions) was investigated by Meijer, his results are
exposed in [3, 30, 31]. Under certain conditions $J(a,b,c,d;u)$ can be
expressed in terms of functions $\Sigma_{\pm}(z)$, where $\Sigma_{+}(z)$ is
the sum of residues of the integrand at poles of factors
$\Gamma(a_{\alpha}+\sigma)$, and $\Sigma_{-}(z)$ the sum of residues at poles
of $\Gamma(b_{\beta}-\sigma)$.

In our case $A=D=q$, $B=C=p$, and $u=|z|$ is a positive real. The contour $L$
coincides with the imaginary axis at infinity. The asymptotics of the absolute
value of the integrand on the imaginary axis is

$\sim|\sigma|^{\sum\limits_{\alpha=1}^{q}\mathop{\mathrm{Re}}\nolimits(a_{\alpha}-d_{\alpha})+\sum\limits_{\beta=1}^{p}\mathop{\mathrm{Re}}\nolimits(b_{\beta}-c_{\beta})},$

see [13, equation (1.18(4))]. One of the statements of [31, Theorem 18]
implies that _if the integrand tends to zero on the imaginary axis, then the
integral $J(a,b,c,d;u)$ conditionally converges if the integrand tends to $0$
at infinity for $u>0$ and is given by_

  * –

$\Sigma_{+}(u)$ for $q>p$;

  * –

$\Sigma_{-}(u)$ for $q<p$;

  * –

for $p=q$ we have $\Sigma_{+}(u)$ if $0<u<1$ and $\Sigma_{-}(u)$ for $u>1$.

If
$\sum\mathop{\mathrm{Re}}\nolimits(a_{\alpha}-d_{\alpha})+\sum\mathop{\mathrm{Re}}\nolimits(b_{\beta}-c_{\beta})<-1$,
then the absolute convergence is obvious.

In our case $a_{\alpha}$ is replaced by $a_{\alpha}+k/2$, $b_{\beta}$ by
$b_{\beta}-k/2$, and $c_{\beta}\mapsto 1-b^{\prime}_{\beta}-k/2$,
$d_{\alpha}\mapsto 1-a^{\prime}_{\alpha}+k/2$. Therefore under our conditions
(1.12), (1.13) all integrals $I_{k}$ converge.

###  2.3 Evaluation of sums of residues

######  Lemma 2.3.

Fix $a|a^{\prime}$ and consider the following family of functions:

$\gamma_{k}(\sigma):=\Gamma^{\mathbb{C}}\big{(}a+\tfrac{k+\sigma}{2}\big{|}a^{\prime}+\tfrac{-k+\sigma}{2}\big{)}.$

Denote by $\Omega$ the sets of all points
$(k,\sigma)\in{\mathbb{Z}}\times{\mathbb{C}}$ such that $\sigma$ is a pole of
$\gamma_{k}(\sigma)$.

  1. $(a)$

The set $\Omega$ is contained in the half-plane
$\mathop{\mathrm{Re}}\nolimits\sigma<0$ if and only if
$\mathop{\mathrm{Re}}\nolimits(a+a^{\prime})>0$.

  2. $(b)$

The set $\Omega$ consists of points

$\displaystyle k=-m+m^{\prime}-a+a^{\prime},$
$\displaystyle\sigma=-m-m^{\prime}-a-a^{\prime},$

where $m$, $m^{\prime}$ range in ${\mathbb{Z}}_{+}$.

###### Proof.

(a) Let all poles be contained in the domain
$\mathop{\mathrm{Re}}\nolimits\sigma<0$. Taking $k=a^{\prime}-a$ we get

$\gamma_{a^{\prime}-a}(\sigma)=\Gamma^{\mathbb{C}}\big{(}a+\tfrac{a^{\prime}-a+\sigma}{2}\big{|}a^{\prime}+\tfrac{-a^{\prime}+a+\sigma}{2}\big{)}=\Gamma^{\mathbb{C}}\big{(}\tfrac{a+a^{\prime}}{2}+\tfrac{\sigma}{2}\big{|}\tfrac{a+a^{\prime}}{2}+\tfrac{\sigma}{2}\big{)}.$

The point $\sigma=-(a+a^{\prime})$ is a pole of
$\gamma_{a^{\prime}-a}(\sigma)$. Therefore,
$\mathop{\mathrm{Re}}\nolimits(a+a^{\prime})>0$.

Conversely, let $\mathop{\mathrm{Re}}\nolimits(a+a^{\prime})>0$. Consider
$(k,\sigma)\in\Omega$. Then (see Section 1.3)

$a+\tfrac{k+\sigma}{2}=-m\in{\mathbb{Z}}_{-},\qquad
a^{\prime}+\tfrac{-k+\sigma}{2}=-m^{\prime}\in{\mathbb{Z}}_{-}.$ (2.2)

Therefore $a+a^{\prime}+\sigma\in{\mathbb{Z}}_{-}$ and
$\mathop{\mathrm{Im}}\nolimits\sigma<0$.

(b) We solve the system (2.2). ∎

###### Evaluation of the sum of residues (the proof of Theorem 1.2).

For definiteness assume that $p\leqslant q$. Let us write the residue
$R(j;m,m^{\prime})$ of the integrand (1.15) at a point

$k=-m+m^{\prime}-a_{j}+a^{\prime}_{j},\qquad\sigma=-m-m^{\prime}-a_{j}-a^{\prime}_{j}.$

We have

$\tfrac{k+\sigma}{2}=-a_{j}-m,\qquad\tfrac{-k+\sigma}{2}=-a_{j}^{\prime}-m^{\prime},$

and keeping in mind (1.9) we get

$\displaystyle
R(j;m,m^{\prime})=\frac{2(-1)^{m}}{m!\,m^{\prime}!}\prod_{\alpha\neq
j}\Gamma^{\mathbb{C}}\big{(}a_{\alpha}-a_{j}-m\big{|}a_{\alpha}^{\prime}-a_{j}^{\prime}-m^{\prime}\big{)}$
$\displaystyle\phantom{R(j;m,m^{\prime})=}{}\times\prod_{\beta}\Gamma^{\mathbb{C}}\big{(}b_{\beta}+a_{j}+m\big{|}b_{\beta}^{\prime}+a_{j}^{\prime}+m^{\prime}\big{)}\times
z^{a_{j}+m|a_{j}^{\prime}+m^{\prime}}.$

Applying (1.8c) and (1.8d) we come to

$\displaystyle
R(j;m,m^{\prime})=2\frac{(-1)^{m}}{m!\,m^{\prime}!}\prod_{\alpha\neq
j}\frac{\Gamma^{\mathbb{C}}\big{(}a_{\alpha}-a_{j}\big{|}a_{\alpha}^{\prime}-a_{j}^{\prime}\big{)}(-1)^{m}}{(1-a_{\alpha}+a_{j})_{m}(1-a_{\alpha}^{\prime}+a_{j}^{\prime})_{m^{\prime}}}$
$\displaystyle\hphantom{R(j;m,m^{\prime})=}{}\times\prod_{\beta}\Gamma^{\mathbb{C}}\big{(}b_{\beta}+a_{j}\big{|}b_{\beta}^{\prime}+a_{j}^{\prime}\big{)}b_{\beta}+a_{j}\big{)}_{m}(b_{\beta}^{\prime}+a_{j}^{\prime})_{m^{\prime}}(-1)^{m^{\prime}}\times
z^{a_{j}+m|a_{j}^{\prime}+m^{\prime}}.$

Reordering factors, we get

$\displaystyle 2z^{a_{j}|a^{\prime}_{j}}\prod_{\alpha\neq
j}\Gamma^{\mathbb{C}}\big{(}a_{\alpha}-a_{j}\big{|}a_{\alpha}^{\prime}-a_{j}^{\prime}\big{)}\prod_{\beta}\Gamma^{\mathbb{C}}\big{(}b_{\beta}+a_{j}\big{|}b_{\beta}^{\prime}+a_{j}^{\prime}\big{)}$
$\displaystyle\qquad{}\times(-1)^{mq}\frac{z^{m}\prod\limits_{\beta}(b_{\beta}+a_{j}\big{)}_{m}}{m!\prod\limits_{\alpha\neq
j}(1-a_{\alpha}+a_{j})_{m}}\times(-1)^{m^{\prime}p}\frac{\overline{z}^{m^{\prime}}\prod\limits_{\beta}(b_{\beta}^{\prime}+a_{j}^{\prime}\big{)}_{m^{\prime}}}{m^{\prime}!\prod\limits_{\alpha\neq
j}(1-a_{\alpha}^{\prime}+a_{j}^{\prime})_{m^{\prime}}}.$

Formal calculation with series gives

$\displaystyle\sum_{m,m^{\prime}}R(j,m,m^{\prime})=2z^{a_{j}|a_{j}^{\prime}}\prod_{\alpha\neq
j}\Gamma^{\mathbb{C}}\big{(}a_{\alpha}-a_{j}\big{|}a_{\alpha}^{\prime}-a_{j}^{\prime}\big{)}\prod_{\beta}\Gamma^{\mathbb{C}}\big{(}b_{\beta}+a_{j}\big{|}b_{\beta}^{\prime}+a_{j}^{\prime}\big{)}$
$\displaystyle\hphantom{\sum_{m,m^{\prime}}R(j,m,m^{\prime})=}{}\times\sum_{m}\frac{\big{(}(-1)^{q}z\big{)}^{m}\prod\limits_{\beta}(b_{\beta}+a_{j}\big{)}_{m}}{m!\prod\limits_{\alpha\neq
j}(1-a_{\alpha}+a_{j})_{m}}\times\sum_{m^{\prime}}\frac{\big{(}(-1)^{p}\overline{z}\big{)}^{m^{\prime}}\prod\limits_{\beta}(b_{\beta}+a_{j}\big{)}_{m^{\prime}}}{m^{\prime}!\prod\limits_{\alpha\neq
j}(1-a_{\alpha}^{\prime}+a_{j}^{\prime})_{m^{\prime}}}.$

We get a product of two hypergeometric series whose radius of convergence is
$\infty$ if $q>p$ and $1$ if $q=p$. They are absolutely convergent in the disk
of convergence, therefore the last identity really takes place. Therefore

$\sum_{j}\sum_{m,m^{\prime}}R(j,m,m^{\prime})=\Sigma^{\mathbb{C}}_{+}(z).$

On the other hand the absolute convergence allows us to write

$\sum_{m,m^{\prime}}R(j,m,m^{\prime})=\sum_{k\in{\mathbb{Z}}}\,\,\sum_{m,m^{\prime}\colon
m-m^{\prime}=k}R(j,m,m^{\prime})=\sum_{k\in{\mathbb{Z}}}(z/\overline{z})^{k/2}I_{k}.$
(2.3)

Thus we proved the coincidence of $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ and
$\Sigma_{+}^{\mathbb{C}}$. The summation of residues at the right poles of
${}_{p}^{\vphantom{{\mathbb{C}}}}\mathcal{K}^{\mathbb{C}}_{q}$ is similar.

It is important that the convergence of the series
$\sum\limits_{k\in{\mathbb{Z}}}$ in (2.3) is locally uniform in the parameters
$a_{\alpha}$, $b_{\beta}$ near any point $(a_{0})\in{\mathbb{C}}^{q}$,
$(b_{0})\in{\mathbb{C}}^{p}$, for which the coefficients at
$z^{l}\overline{z}^{l^{\prime}}$ are well-defined. ∎

###  2.4 The application of the Mellin transform

Our next purpose is Theorem 1.7 about evaluations of the convolution integrals

$\int_{{\mathbb{C}}}\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G(\dots;z)\,\fourIdx{}{r}{{\mathbb{C}}}{\\!s}G(\dots;t/z)\,\frac{\,{\rm
d}\,{\overline{\overline{z}}}}{z^{1|1}}.$

Under the condition of Corollary 2.1 the integral (1.15) defining a
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$-function is the inverse Mellin transform
of the function $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}$. Therefore
the function $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}{\mathcal{K}}$ is the direct
Mellin transform of $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$. This is the
statement of Lemma 1.6.

The Mellin transform is the Fourier transform on the group
${\mathbb{C}}^{\times}$, therefore it sends multiplicative convolutions to
products. This property remains to be valid if both functions are contained in
$L^{2}$, see [44, Theorem 64, Section 3.13].444Proof. Denote by $C_{0}$ the
space of continuous functions that have zero limit at infinity. The product is
a continuous operation $L^{2}\times L^{2}\to L^{1}$. The Fourier transform
sends $L^{2}\to L^{2}$ bijectively, and sends $L^{1}\to C_{0}$. The
convolution is a continuous operation $L^{2}\times L^{2}\to C_{0}$. Therefore
the Fourier transform sends a product of two $L^{2}$-functions to the
convolution. This implies Theorem 1.7.

###  2.5 The differential equations

Here we derive the system (1.19) of differential equations for a function
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ (the statement of Corollary 1.4). We can
use the explicit expressions $\Sigma_{\pm}^{\mathbb{C}}$ for
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ obtained in Theorem 1.2. The formula
(1.18) for $\Sigma_{+}^{\mathbb{C}}$ has the form

$\sum C_{j}\Phi_{j}(z)\cdot\Psi_{j}(\overline{z}),$

where $C_{j}$ do not depend on $z$ and

$\Phi_{j}(z)=z^{a_{j}}\fourIdx{}{p}{}{\\!q-1}F\left[\genfrac{}{}{0.0pt}{}{(b_{\alpha}+a_{j})}{(1-a_{\alpha}+a_{j})_{\setminus
j}};(-1)^{p+q}z\right].$

The equation $\mathcal{D}F=0$ of the system (1.19) is slightly modified
equation for generalized hypergeometric functions, see [43, Section 2.1.2] or
[30, Section 5.2]. The functions $\Phi_{j}(z)$ form a fundamental system of
solutions of this equation in holomorphic functions on ${\mathbb{C}}$.

This implies Corollary 1.4.

######  Remark 2.4.

The statement also follows from the identities (3.10a)–(3.11b) established
below.

###  2.6 The analytic continuation

It remains to prove Proposition 1.5 about the analytic continuation of the
function
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\big{[}\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\big{]}$
in the variable $z$ and in the parameters $(a|a^{\prime})$, $(b|b^{\prime})$.

(1). Assume $p\neq q$, for definiteness let $p<q$. First, we omit the
condition (1.12). A value of $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ at $z$ is
defined by an infinite sum of contour integrals $I_{k}$ and the convergence is
locally uniform in the parameters. For all but a finite number of summands the
imaginary axis ${\rm i}{\mathbb{R}}$ separates the left poles and right poles
of the integrand (see the terminology of Section 1.4). Consider one of the
remaining summands, say $I_{m}$. A contour $L_{m}$ separating left poles and
right poles of (1.15) exists if and only if left poles differ from right
poles. The condition (1.21) is the condition of collision of left and right
poles. Two separation contours $L_{m}$, $L_{m}^{\prime}$ can be nonhomotopic.
However

$\frac{1}{2\pi{\rm i}}\left(\int_{L_{m}}-\int_{{\rm i}{\mathbb{R}}}\right)$

is a sum of residues at right poles contained in the half-plane
$\mathop{\mathrm{Re}}\nolimits\sigma<0$ minus a sum of residues at left poles
contained in the half-plane $\mathop{\mathrm{Re}}\nolimits\sigma>0$. This
expression does not depend on a choice of separation contour. Clearly, a
contour integral is holomorphic in the parameters.

So the sum of contour integrals is well-defined in the domain defined by
conditions (1.16), (1.17) (absolute convergence and absence of collisions).

On the other hand the generalized hypergeometric functions
$\fourIdx{}{p}{}{\\!q-1}F$ meromorphically depend on their parameters, and
therefore $\Sigma_{+}(z)$ meromorphically depends on parameters and this
allows to omit the condition (1.13) for the convergence. Sums of power series
are real analytic and this implies real analyticity of $\Sigma_{+}(z)$ for
$z\in{\mathbb{C}}\setminus 0$.

Possible poles are located at hyperplanes

$\begin{cases}a_{j}+b_{\beta}\in{\mathbb{Z}}_{-},\\\
a_{j}^{\prime}+b_{\beta}^{\prime}\in{\mathbb{Z}}_{-},\end{cases}$

for some $j$ and $\beta$ and

$\begin{cases}a_{\alpha}-a_{j}\in{\mathbb{Z}},\\\
a_{\alpha}^{\prime}-a_{j}^{\prime}\in{\mathbb{Z}},\end{cases}$

for some $j$, $\alpha$. It remains to notice that singularities of the second
type are removable under the open condition (1.16) and therefore they are
always removable.

(2). Let $p=q$. The same argument can be applied for $|z|<1$ and $|z|>1$, it
remains to examine the function on the circle $|z|=1$. We extend
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G(z)$ to the same domain of parameters as
above and come away to a smaller domain555So we assume $p>2$. For the case
$p=2$ the analyticity on the circle is clear from explicit formulas, see [33];
the case $p=1$ is trivial, see (3.2) below.

$\sum\mathop{\mathrm{Re}}\nolimits(a_{\alpha}+a_{\alpha}^{\prime})+\sum\mathop{\mathrm{Re}}\nolimits(b_{\beta}+b_{\beta}^{\prime})<p-2.$
(2.4)

Then the contour integral in (1.15) has continuous partial derivatives up to
order $p$. By continuity the equation $\mathcal{D}F=0$ is valid on the circle
$|z|=1$. The operator $\mathcal{D}$ is elliptic for $z\neq 0$, $\pm 1$.
Therefore under the condition (2.4) solutions of the equation $\mathcal{D}F=0$
are analytic (see, e.g., [22, Theorem 8.5.1]). Consider a point $z_{0}\neq 1$
on the circle $|z|=1$. Any generalized hypergeometric function defined in the
disk $|z|<1$ has an analytic continuation to a neighborhood $U$ of $z_{0}$.
Therefore the expression (1.18) for $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G(z)$
provides us an analytic continuation of $\Sigma_{+}(z)$ to $U$ (and coincides
with $\Sigma_{-}^{\mathbb{C}}$ in the intersection of $U$ and the domain
$|z|>1$). This expression is meromorphic in $(a)$, $(b)$ as above.

##  3 Additions

###  3.1 Some simple cases

####  3.1.1 The exponential

For $p=0$, $q=1$ formula (1.18) gives

$\fourIdx{}{0}{{\mathbb{C}}}{\\!1}G\left[\genfrac{}{}{0.0pt}{}{a|a^{\prime}}{\text{--}};z\right]=2z^{a|a^{\prime}}{\rm
e}^{-z}{\rm e}^{\overline{z}}.$ (3.1)

####  3.1.2 The power function

For $p=q=1$ we get

$\displaystyle\fourIdx{}{1}{{\mathbb{C}}}{\\!1}G\left[\genfrac{}{}{0.0pt}{}{a|a^{\prime}}{b|b^{\prime}};z\right]$
$\displaystyle=2z^{a|a^{\prime}}\Gamma^{\mathbb{C}}(a+b|a^{\prime}+b^{\prime})\fourIdx{}{1}{}{\\!0}F\left[\genfrac{}{}{0.0pt}{}{a+b}{\text{--}};-z\right]\cdot\fourIdx{}{1}{}{\\!0}F\left[\genfrac{}{}{0.0pt}{}{a^{\prime}+b^{\prime}}{\text{--}};-\overline{z}\right]$
$\displaystyle=2\Gamma^{\mathbb{C}}(a+b|a^{\prime}+b^{\prime})\,z^{a|a^{\prime}}(1+z)^{-a-b|-a^{\prime}-b^{\prime}}.$
(3.2)

The series $\fourIdx{}{1}{}{\\!0}F$ converge in the disc $|z|<1$, but the
final expression is well defined in ${\mathbb{C}}\setminus\\{0,-1\\}$.

####  3.1.3 The Kummer functions

For $p=1$, $q=2$ we get

$\displaystyle\fourIdx{}{1}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b|b^{\prime}};z\right]=2\Gamma^{\mathbb{C}}(a_{1}+b|a_{1}^{\prime}+b)\,\Gamma^{\mathbb{C}}(a_{2}-a_{1}|a_{2}^{\prime}-a_{1}^{\prime})$
$\displaystyle\hphantom{\fourIdx{}{1}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b|b^{\prime}};z\right]=}{}\times
z^{a_{1}}\fourIdx{}{1}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a_{1}+b}{1-a_{2}+a_{1}};z\right]\overline{z}^{a_{1}^{\prime}}\fourIdx{}{1}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a_{1}^{\prime}+b^{\prime}}{1-a_{2}^{\prime}+a_{1}^{\prime}};-\overline{z}\right]$
(3.3)
$\displaystyle\hphantom{\fourIdx{}{1}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b|b^{\prime}};z\right]=}{}+\big{\\{}\text{similar
term obtained by the transposition $a_{1}|a_{1}^{\prime}\longleftrightarrow
a_{2}|a_{2}^{\prime}$}\big{\\}}.$

Denote the hypergeometric functions in these expression by $\Phi_{1}(z)$,
$\Phi_{1}^{\prime}(\overline{z})$, $\Phi_{2}(z)$,
$\Phi_{2}^{\prime}(\overline{z})$. Then $z^{a_{1}}\Phi_{1}(z)$,
$z^{a_{2}}\Phi_{2}(z)$ is a fundamental system of holomorphic solutions of the
equation $\mathcal{D}F=0$ of the system (1.19), and
$\overline{z}^{a_{1}^{\prime}}\Phi_{1}^{\prime}(\overline{z})$,
$\overline{z}^{a_{2}^{\prime}}\Phi_{2}^{\prime}(\overline{z})$ is a
fundamental system of antiholomorphic solutions of the equation
$\overline{\mathcal{D}}F=0$ (generally, all these functions are ramified at
$0$ and $\infty$). Then the functions

$z^{a_{1}}\Phi_{1}(z)\,\overline{z}^{a_{1}^{\prime}}\Phi_{1}^{\prime}(\overline{z}),\quad
z^{a_{2}}\Phi_{2}(z)\,\overline{z}^{a_{1}^{\prime}}\Phi_{1}^{\prime}(\overline{z}),\quad
z^{a_{1}}\Phi_{1}(z)\,\overline{z}^{a_{2}^{\prime}}\Phi_{2}^{\prime}(\overline{z}),\quad
z^{a_{2}}\Phi_{2}(z)\,\overline{z}^{a_{2}^{\prime}}\Phi_{2}^{\prime}(\overline{z})$

is a basis of the space of solutions of the system (1.19) in a neighborhood of
any point $z_{0}\neq 0$, see [33, Proposition 3.8]. For
$a_{1}-a_{2}\notin{\mathbb{Z}}$ solutions non-ramified at $0$ and $\infty$
have the form

$C_{1}\,z^{a_{1}|a_{1}^{\prime}}\Phi_{1}(z)\Phi_{1}^{\prime}(\overline{z})+C_{2}\,z^{a_{2}|a_{2}^{\prime}}\Phi_{2}(z)\Phi_{2}^{\prime}(\overline{z}).$
(3.4)

The asymptotic expansion of confluent hypergeometric function
$\fourIdx{}{1}{}{\\!1}F(z)$ as $z\to\infty$ (see, e.g., [30, Section 4.7]) in
the sector $|\arg z|<\pi-\varepsilon$ is

$\displaystyle\fourIdx{}{1}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a}{b};z\right]=\frac{\Gamma(b)}{\Gamma(b-a)}{\rm
e}^{{\rm i}\pi\mathop{\mathrm{sgn}}\nolimits(\mathop{\mathrm{Im}}\nolimits
a)}z^{-a}\left(\sum_{n=0}^{K-1}\frac{(a)_{n}(1+a-b)_{n}}{n!}(-z)^{-n}+O\big{(}z^{-K}\big{)}\right)$
$\displaystyle\hphantom{\fourIdx{}{1}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{a}{b};z\right]=}{}+\frac{\Gamma(b)}{\Gamma(a)}e^{z}z^{a-b}\left(\sum_{n=0}^{L-1}\frac{(b-a)_{n}(1-a)_{n}}{n!}z^{-n}+O\big{(}z^{-L}\big{)}\right).$

For generic $C_{1}$, $C_{2}$ the growth of expression (3.4) as
$|\mathop{\mathrm{Re}}\nolimits z|\to\infty$ is exponential. For the linear
combination (3.3) all exponential terms of the asymptotics disappear (and
ratio of coefficients in (3.3) is uniquely defined by this condition,
existence of such ratio a priori is non-obvious). A reminder is
$O\big{(}z^{-N}{\rm e}^{z}\big{)}+O\big{(}z^{-N}{\rm e}^{-z}\big{)}+O(1)$ for
any $N$, is too rough. However, under the conditions
$\mathop{\mathrm{Re}}\nolimits(a_{1}+a_{1}^{\prime})>0$,
$\mathop{\mathrm{Re}}\nolimits(a_{2}+a_{2}^{\prime})>0$,
$\mathop{\mathrm{Re}}\nolimits(a_{1}+a_{1}^{\prime}+a_{2}+a_{2}^{\prime})<1$
our function $\fourIdx{}{2}{{\mathbb{C}}}{\\!1}G$ is contained in
$L^{2}\big{(}{\mathbb{C}},|z|^{-2}\,{\rm
d}\,{\overline{\overline{z}}}\big{)}$.

####  3.1.4 The Bessel functions

For $p=0$, $q=2$ we get an expression

$\displaystyle\fourIdx{}{0}{{\mathbb{C}}}{\\!2}G\\!\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{\text{--}};z\right]=2\Gamma^{\mathbb{C}}(a_{2}-a_{1}|a_{2}^{\prime}-a_{1}^{\prime})z^{a_{1}|a_{1}^{\prime}}\,\fourIdx{}{0}{}{\\!1}F\\!\left[\genfrac{}{}{0.0pt}{}{\text{--}}{1+a_{1}-a_{2}};z\right]\fourIdx{}{0}{}{\\!1}F\\!\left[\genfrac{}{}{0.0pt}{}{\text{--}}{1+a_{1}^{\prime}-a_{2}^{\prime}};\overline{z}\right]$
$\displaystyle\hphantom{\fourIdx{}{0}{{\mathbb{C}}}{\\!2}G\\!\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{\text{--}};z\right]=}{}+\big{\\{}\text{similar
term obtained by the transposition $a_{1}|a_{1}^{\prime}\longleftrightarrow
a_{2}|a_{2}^{\prime}$}\big{\\}}.$

####  3.1.5 The Gauss hypergeometric functions

For $p=q=2$

$\displaystyle\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b_{1}|b_{1}^{\prime},b_{1}|b_{1}};z\right]=z^{a_{1}|a_{1}^{\prime}}\Gamma^{\mathbb{C}}(a_{2}-a_{1}|a_{2}^{\prime}-a_{1}^{\prime})\prod_{\beta=1}^{2}\Gamma^{\mathbb{C}}(b_{\beta}+a_{1}|b_{\beta}^{\prime}+a_{1}^{\prime})$
$\displaystyle\hphantom{\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b_{1}|b_{1}^{\prime},b_{1}|b_{1}};z\right]=}{}\times\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{b_{1}+a_{1},b_{2}+a_{1}}{1+a_{1}-a_{2}};z\right]\,\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{b_{1}^{\prime}+a_{1}^{\prime},b_{2}^{\prime}+a_{1}^{\prime}}{1+a_{1}^{\prime}-a_{2}^{\prime}};z\right]$
$\displaystyle\hphantom{\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G\left[\genfrac{}{}{0.0pt}{}{a_{1}|a_{1}^{\prime},a_{2}|a_{2}^{\prime}}{b_{1}|b_{1}^{\prime},b_{1}|b_{1}};z\right]=}{}+\big{\\{}\text{similar
term obtained by the transposition $a_{1}|a_{1}^{\prime}\longleftrightarrow
a_{2}|a_{2}^{\prime}$}\big{\\}}.$

On the other hand Gelfand, Graev and Retakh [16] defined the analog of the
Gauss hypergeometric function by the Euler integral (see the detailed
examination in [33, Section 3]):

$\displaystyle\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{A|A^{\prime},B|B^{\prime}}{C|C^{\prime}};z\right]:=\frac{\Gamma^{{\mathbb{C}}}(B|B^{\prime})}{\Gamma^{{\mathbb{C}}}(B|B^{\prime})\,\Gamma^{{\mathbb{C}}}(C-B|C^{\prime}-B^{\prime})}$
$\displaystyle\hphantom{\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{A|A^{\prime},B|B^{\prime}}{C|C^{\prime}};z\right]:=}{}\times\int_{\mathbb{C}}t^{B-1|B^{\prime}-1}(1-t)^{C-B-1|C^{\prime}-B^{\prime}-1}(1-zt)^{-A|-A^{\prime}}\,{\rm
d}\,{\overline{\overline{t}}}.$ (3.5)

We have

$\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{A|A^{\prime},B|B^{\prime}}{C|C^{\prime}};z\right]=\frac{\Gamma(C|C^{\prime})(-1)^{C-C^{\prime}}}{\Gamma(A|A^{\prime})\,\Gamma(B|B^{\prime})}\,\,\fourIdx{}{2}{}{\\!2}F\left[\genfrac{}{}{0.0pt}{}{0|0,1-C|1-C^{\prime}}{A|A^{\prime},B|B^{\prime}};z\right].$

We do not know which notation is better. In any case, for the notation
$\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F$ formulas are precisely parallel to the
classical theory of the Gauss hypergeometric functions.

As for the Kummer and Bessel cases the system (1.19) has $4$-dimensional space
of local solutions and a two-dimensional subspace of solutions that are non-
ramified at $0$. The function $\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F$ is selected
from this subspace by the condition of non-ramification at $z=1$, see [33,
Proposition 3.11].

###  3.2 Some simple properties of the functions
$\boldsymbol{\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G}$

Here we imitate properties of the Meijer $G$-function (see Prudnikov, Brychkov
and Marichev [40, Vol. 3, Section 8.2]). Clearly, the function
$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ is symmetric with respect to permutations
of $a_{1}|a_{1}^{\prime},\dots,a_{p}|a_{p}^{\prime}$ and of
$b_{1}|b_{1}^{\prime},\dots,b_{q}|b_{q}^{\prime}$. If
$a_{m}-a_{l}\in{\mathbb{Z}}$, then we have an additional symmetry

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
m,l},a_{m}|a_{l}^{\prime},a_{m}^{\prime}|a_{l}}{(b|b^{\prime})};z\right],\qquad\text{if
$a_{k}-a_{l}\in{\mathbb{Z}}$}$

(if $a_{k}-a_{l}\notin{\mathbb{Z}}$, then the right hand side makes no sense),
this follows from (1.7).

Changing the variables $k\mapsto-k$ and $\sigma\mapsto-\sigma$ in (1.15) we
get

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\fourIdx{}{q}{{\mathbb{C}}}{\\!p}G\left[\genfrac{}{}{0.0pt}{}{(b|b^{\prime})}{(a|a^{\prime})};z^{-1}\right].$
(3.6)

Changing only the summation index $k\mapsto-k$, we get

$\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=(-1)^{\sum(a_{\alpha}-a_{\alpha}^{\prime})+\sum(b_{\beta}-b_{\beta}^{\prime})}\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};\overline{z}\right].$

Keeping in the mind the reflection formula (1.8b), we get

$\fourIdx{}{p+1}{{\mathbb{C}}}{\\!q+1}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime}),}{(b|b^{\prime}),}\genfrac{}{}{0.0pt}{}{c|c^{\prime}}{1-c|1-c^{\prime}};z\right]=(-1)^{c-c^{\prime}}\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right].$
(3.7)

Shifting variables $k\mapsto k+l$, $\sigma\mapsto\sigma+\tau$, we come to

$z^{c|c^{\prime}}\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})+c|c^{\prime}}{(b|b^{\prime})-c|c^{\prime}};z\right].$
(3.8)

Keeping in mind (1.8b), we obtain

$\displaystyle\fourIdx{}{mp}{{\mathbb{C}}}{\\!mq}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime}),(a|a^{\prime})+\frac{1}{m}|\frac{1}{m},\dots,(a|a^{\prime})+\frac{m-1}{m}|\frac{m-1}{m}}{(b|b^{\prime}),(b|b^{\prime})+\frac{1}{m}|\frac{1^{\vphantom{R}}}{m},\dots,(b|b^{\prime})+\frac{m-1}{m}|\frac{m-1}{m}};z\right]$
$\displaystyle\qquad{}=m^{p+q-2-\sum(a_{\alpha}+a_{\alpha}^{\prime})-\sum(b_{\beta}+b_{\beta}^{\prime})}\sum_{l=0}^{m-1}\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(ma|ma^{\prime})}{(mb|mb^{\prime})};{\rm
e}^{2\pi{\rm i}l/m}z^{1/m}m^{mp(-q)}\right],$ (3.9)

here we have a summation, which is absent for the classical Meijer
$G$-function.

Differentiating the integral by the parameter $z$ we get

$\displaystyle\left({-}z\frac{\partial}{\partial
z}+a_{j}\right)\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
j},(a_{j}+1)|a_{j}^{\prime}}{(b|b^{\prime})};z\right],$ (3.10a)
$\displaystyle\left(z\frac{\partial}{\partial
z}+b_{m}\right)\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})_{\setminus
m},(b_{m}+1)|b_{m}^{\prime}};z\right].$ (3.10b)

Due to the $(-1)$ in (1.8c), the similar equations for
$\overline{z}\frac{\partial}{\partial\overline{z}}$ differ from (3.10a) and
(3.10b) by signs

$\displaystyle\left({-}\overline{z}\frac{\partial}{\partial\overline{z}}+a_{j}^{\prime}\right)\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=-\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
j},a_{j}|(a_{j}^{\prime}+1)}{(b|b^{\prime})};z\right],$ (3.11a)
$\displaystyle\left(\overline{z}\frac{\partial}{\partial\overline{z}}+b^{\prime}_{m}\right)\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right]=-\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})_{\setminus
m},b_{m}|(b_{m}^{\prime}+1)};z\right].$ (3.11b)

The last four equalities (3.10a)–(3.11b) imply the differential equations
(1.19)–(1.20b).

Also

$\displaystyle\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
j},(a_{j}+1)|a_{j}^{\prime}}{(b|b^{\prime})};z\right]-\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
m},(a_{m}+1)|a_{m}^{\prime}}{(b|b^{\prime})};z\right]$
$\displaystyle\qquad{}=(a_{j}-a_{m})\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right],$

and

$\displaystyle\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})_{\setminus
j},(a_{j}+1)|a_{j}^{\prime}}{(b|b^{\prime})};z\right]+\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})_{\setminus
m},(b_{m}+1)|b_{m}^{\prime}};z\right]$
$\displaystyle\qquad{}=(a_{j}+b_{m})\,\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G\left[\genfrac{}{}{0.0pt}{}{(a|a^{\prime})}{(b|b^{\prime})};z\right].$

###  3.3 References, links and problems

(1). Gauss hypergeometric functions of the complex field (our
$\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G$ modulo a simple factor) were defined by
Gelfand, Graev and Retakh in [16] by the Euler integral (3.5). Many formulas
for such functions were obtained in [33, Section 3].

(2). Marichev [31] proposed the following trick, which became the main tool in
the creation of the Prudnikov, Brychkov and Marichev tables [40],666See also
the tables [4]. see also comments in [41]. Many special functions (and many
elementary functions) are special cases of the Meijer $G$-functions, i.e., can
be written as Mellin–Barnes integrals (2.1). Therefore they are inverse Mellin
transforms of products

$\frac{\prod\limits_{\alpha=1}^{A}\Gamma(A_{\alpha}+\sigma)\prod\limits_{\beta=1}^{B}\Gamma(B_{\beta}-\sigma)}{\prod\limits_{\gamma=1}^{C}\Gamma(C_{\gamma}+\sigma)\prod\limits_{\delta=1}^{D}\Gamma(D_{\delta}-\sigma)}.$
(3.12)

Take two such functions $\Phi(x)$, $\Psi(x)$. Then we can evaluate the
convolution

$\Theta(x)=\int_{0}^{\infty}\Phi(y)\Psi(x/y)y^{-1}\,{\rm d}y.$

Indeed, the Mellin transform of $\Theta$ is the product of Mellin transforms,
therefore we get a product of two functions of the type (3.12), i.e., a
function of the same type. Now we can express $\Theta(x)$ as a linear
combination of hypergeometric functions.

Numerous formulas in tables of integrals (such as Gradshteyn and Ryzhik [18])
whose initial derivations were ingenious can be obtained in this
straightforward way. The table of evaluations of $G$-functions in Prudnikov,
Brychkov and Marichev [40, Vol. 3, Section 8.4] contains 90 pages, for each
pair of lines we can apply this trick.777Chapter 7 of the same book (160
pages) also provides us a material for this business.

Our Theorem 1.7 with formulas (3.6), (3.7), (3.8) gives us the same
tool.888Our arguments are not sufficient for integrals (1.22) with functions
$\fourIdx{}{1}{{\mathbb{C}}}{\\!0}G$, see (3.1). Apparently, the formula
(1.22) remains valid in this case. However, in our case the picture is less
sophisticated. The classical Meijer functions depend on $4$ subscripts and
superscripts (see (2.1)), In our case the reflection formula (1.8b) allows to
move $\Gamma^{\mathbb{C}}$-factors from the denominator to numerator. As a
result, functions $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$ depend only on two
subscripts $p$, $q$.

Apparently, most999With some exceptions, for instance an application of
formula (3.9) can be problematic. of identities for classical hypergeometric
functions as they are exposed in [13, 14] (Chapters 2, 4, 6, 7), [1, 43] have
counterparts for functions $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$, but different
classical formulas can have the same counterpart (for instance the
${{}_{5}H_{5}}$-Dougall formula and the de Branges–Wilson integral correspond
to one integral over ${\mathbb{Z}}\times{\mathbb{R}}$, see [39]).

(3). A collection of beta-integrals involving products of
$\Gamma^{\mathbb{C}}$ is known, see Bazhanov, Mangazeev and Segeev [2], Kels
[27, 28], Derkachov, Manashov and Valinevich [6, 8, 9, 10], Neretin [39],
Sarkissian and Spiridonov [42], in particular, this collection contains
counterparts of the de Branges–Wilson integral and the Nassrallah–Rahman
integral. Such integrals can be regarded as evaluations of functions
$\fourIdx{}{p}{{\mathbb{C}}}{\\!p}G(z)$ at the point $(-1)^{p}$.

(4). Such beta integrals and such hypergeometric functions arise as limits of
elliptic beta-integrals and the hypergeometric functions
$\fourIdx{}{p}{{\mathbb{C}}}{\\!p}G$ as limits of elliptic hypergeometric
functions, see Sarkissian and Spiridonov [42].

(5). The classical expansion in Jacobi polynomials has a well-known continuous
analog known under terms ‘Olevski transform’, ‘generalized Mehler–Fock
transform’, ‘Jacobi transform’, see, e.g., [29]; there is also a second
continuous analog [38]. The paper Molchanov and Neretin [33] contains a
complex counterpart of these $3$ transformations (expansions in the Jacobi
polynomials and two integral operators), it is a unitary integral transform
with $\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G$-kernel acting from a certain
weighted $L^{2}$ on ${\mathbb{C}}$ to a certain weighted $L^{2}$ on
$\Lambda\simeq{\mathbb{Z}}\times{\mathbb{R}}$.

The ‘Jacobi transform’ is a representative of a big zoo of hypergeometric
integral transforms (see, e.g., [46]), it is natural to think that their
counterparts exist in $\fourIdx{}{p}{{\mathbb{C}}}{\\!q}G$-cases. Integral
transforms also can be applied as a tool for the production of special-
functional identities (clearly, several transforms were used for the creation
of [40, Vol. 3], on possibilities of the Jacobi transform, see [37]). An
example of application of the $\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G$-transform
is contained in [39].

(6). It is well-known that representation theory of the group
$\mathrm{SL}(2,{\mathbb{R}})$ is closely related to theory of hypergeometric
functions (as the Bessel functions, the confluent hypergeometric functions,
the Gauss hypergeometric functions, $\fourIdx{}{3}{}{\\!2}F(1)$, and the
balanced $\fourIdx{}{4}{}{\\!3}F(1)$). Application of harmonic analysis
related to the Lorentz group101010See old works Gelfand, Graev and Vilenkin
[17], Naimark [34, 35, 36], Gelfand and Graev [15] on
$\mathrm{SL}(2,{\mathbb{C}})$-harmonic analysis, see also Derkachov,
Korchemsky and Manashov [5, 7]. $\mathrm{SL}(2,{\mathbb{C}})$ to special
functions are far not so popular (at least among pure mathematicians).
However, if to ask such a question, then hypergeometric functions of complex
field come thick and fast.

— a tensor product $\rho_{1}\otimes\rho_{2}$ of two irreducible unitary
representations of $\mathrm{SL}(2,{\mathbb{C}})$ is a multiplicity free direct
integral (see Naimark [34, 35, 36]). Therefore we can canonically decompose a
triple tensor product

$(\rho_{1}\otimes\rho_{2})\otimes\rho_{3}=\rho_{1}\otimes(\rho_{2}\otimes\rho_{3})$

in two ways. In the first case we decompose $\rho_{1}\otimes\rho_{2}$ and
multiply each component by $\rho_{3}$, in the second case we start from
$\rho_{2}\otimes\rho_{3}$. Realizing this idea111111It is difficult to extend
this approach to unitary representations of other groups since decompositions
of tensor products usually have multiplicities $>1$ (even for
$\mathrm{SL}(2,{\mathbb{R}})$). Several multiplicity free cases were examined
in [20, 23, 24, 25, 26]. we get two explicit decompositions of the same
representation, the intertwining operator splits into a direct integral of
intertwining operators acting in isotypic components, such operators can be
regarded as counterparts of Racah coefficients ($6j$-symbols). Ismagilov [23,
24] (see also Derkachov iand Spiridonov [11]) showed that such ‘Racah
operators’ are integrals transforms whose kernels have a form
$\fourIdx{}{4}{{\mathbb{C}}}{\\!4}G(1)$. The ‘Racah operators’ are
$G^{\mathbb{C}}$-counterparts of expansions in Racah polynomials, expansions
in Wilson polynomials, and the ‘Wilson function transforms’ defined by
Groenevelt [19, 20].

— Recall that the Lorentz group $\mathrm{SL}(2,{\mathbb{C}})$ is locally
isomorphic to the complex orthogonal group $\mathrm{SO}(3,{\mathbb{C}})$.
Consider the symmetric space
$\mathrm{SO}(3,{\mathbb{C}})/\mathrm{SO}(2,{\mathbb{C}})$, it can be regarded
as a quadric $x^{2}+y^{2}+z^{2}=1$ in ${\mathbb{C}}^{3}$, or the
complexification of the sphere $x^{2}+y^{2}+z^{2}=1$ in ${\mathbb{R}}^{3}$.
Under the complexification, the Laplacian on the real sphere splits into two
commuting Laplacians, one is holomorphic, another is anti-holomorphic. A
question about their joint spectral decomposition in a space of
$\mathrm{SO}(2)$-invariant functions leads to ${{}_{2}G_{2}}$-transform
considered in Molchanov and Neretin [33].

The shortest way of appearance of
$\fourIdx{}{2}{{\mathbb{C}}}{\\!2}G$-functions is discussed in Section 3.4
below.

(7). Dotsenko and Fateev [12] obtained a complex version of the Selberg
integral; Derkachov, Manashov and Valinevich [9, 10] obtained multi-
dimensional beta-integrals with products of $\Gamma^{\mathbb{C}}$-functions
(counterparts of the Gustafson’s extension of the second Barnes lemma).

There arises a question about multi-dimensional symmetric
${\mathbb{C}}$-counterparts of the Heckman–Opdam hypergeometric functions
[21]. The obvious candidates are spherical distributions on symmetric spaces
$G_{\mathbb{C}}/K_{\mathbb{C}}$, where $G_{\mathbb{C}}$ is a complex
semisimple Lie group and $K_{\mathbb{C}}$ a complex symmetric subgroup.

On the other hand there are no reasons to hope that in the multi-dimensional
case complex spherical transforms are unitary operators (unexpectedly, the
radial parts of Laplace operators can be non-commuting in the Nelson sense,
see [33, Theorem 1.1]).

###  3.4 The Vilenkin model for $\boldsymbol{\mathrm{SL}(2,{\mathbb{C}})}$

First, we modify notation. Let us denote elements of $\Lambda_{\mathbb{C}}$ by
bold letters, denote $a|a^{\prime}$ by $\mathbf{a}$, $1|1$ by
$\boldsymbol{1}$, and $(-1)^{a-a^{\prime}}$ by $(-1)^{\mathbf{a}}$. Since
$\Lambda\simeq{\mathbb{Z}}\times{\mathbb{R}}$, we can denote

$\int_{\Lambda}(\dots)=\sum_{k\in{\mathbb{Z}}}\int_{\mathbb{R}}(\dots).$

Now let us explain how the hypergeometric functions
$\fourIdx{}{2}{{\mathbb{C}}}{\\!1}F$ arise from representations of the Lorentz
group $\mathrm{SL}(2,{\mathbb{C}})$. Recall that this group can be realized as
the group of all complex $2\times 2$ matrices
$\big{(}\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\big{)}$ with
determinant $ad-bc=1$. The _principal series of representations_
$T_{\boldsymbol{\sigma}}$ of this group is parametrized by
${\boldsymbol{\sigma}}=\sigma|\sigma^{\prime}\in\Lambda_{\mathbb{C}}$. They
act in the space of functions on ${\mathbb{C}}$ by operators

$T_{\boldsymbol{\sigma}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}f(z):=f\left(\frac{b+zd}{a+zc}\right)(a+zc)^{-\boldsymbol{1}+{\boldsymbol{\sigma}}}.$

For $\sigma\in\Lambda$ we get unitary representations in
$L^{2}({\mathbb{C}})$, for details, see [17, Chapter III].

Let us realize the representations $T_{\boldsymbol{\sigma}}$ in a space of
functions on $\Lambda$ conjugating them by the Mellin transform121212Cf. a
model of representations of $\mathrm{SL}(2,{\mathbb{R}})$ in Vilenkin [45,
Section VII.3].

$\widetilde{T}_{\boldsymbol{\sigma}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\mathcal{M}\circ T_{\boldsymbol{\sigma}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\circ\mathcal{M}^{-1}.$

A straightforward calculation shows that

$\widetilde{T}_{\boldsymbol{\sigma}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}F(\mu)=\frac{1}{4\pi^{2}{\rm
i}}\int_{\Lambda}\mathscr{L}\left[{\boldsymbol{\mu}},{\boldsymbol{\lambda}};\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right]\,F({\boldsymbol{\lambda}})\,{\rm
d}{\boldsymbol{\lambda}},$

where

$\mathscr{L}\left[{\boldsymbol{\mu}},{\boldsymbol{\lambda}};\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right]=\int_{\mathbb{C}}z^{{\boldsymbol{\mu}}-1}(a+zc)^{{\boldsymbol{\sigma}}-{\boldsymbol{\lambda}}-1}(b+zd)^{{\boldsymbol{\lambda}}}\,\,{\rm
d}\,{\overline{\overline{z}}}.$

Substituting $z=-\frac{a}{c}u$ we come to

$\mathscr{L}\left[{\boldsymbol{\mu}},{\boldsymbol{\lambda}};\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right]=(-1)^{\boldsymbol{\mu}}a^{{\boldsymbol{\sigma}}+{\boldsymbol{\mu}}-{\boldsymbol{\lambda}}-\boldsymbol{1}}b^{\boldsymbol{\lambda}}c^{-{\boldsymbol{\mu}}}\,\frac{\Gamma^{\mathbb{C}}({\boldsymbol{\sigma}}-{\boldsymbol{\lambda}}+{\boldsymbol{\mu}})}{\Gamma^{\mathbb{C}}({\boldsymbol{\mu}})\Gamma^{\mathbb{C}}({\boldsymbol{\sigma}}-{\boldsymbol{\lambda}})}\,\,\fourIdx{}{2}{}{\\!1}F\left[\genfrac{}{}{0.0pt}{}{{\boldsymbol{\mu}},-{\boldsymbol{\lambda}}}{{\boldsymbol{\sigma}}-{\boldsymbol{\lambda}}+{\boldsymbol{\mu}}};\frac{ad}{bc}\right].$

### Acknowledgements

The work was supported by the grant FWF, P31591. I am grateful to M.I. Graev,
V.F. Molchanov, V.A. Spiridonov, and S.É. Derkachov for discussions and
references, I also thank the referees for useful comments.

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