Transmutation Operators

Fractional Integral and Generalized Stieltjes
Transforms for Hypergeometric Functions
as Transmutation Operators\star\star$\star$This paper is a contribution to the
Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc
Vinet. The full collection is available at
http://www.emis.de/journals/SIGMA/ESSA2014.html

Tom H. KOORNWINDER

T.H. Koornwinder

Korteweg-de Vries Institute, University of Amsterdam,
P.O. Box 94248, 1090 GE Amsterdam, The Netherlands<EMAIL_ADDRESS>https://staff.fnwi.uva.nl/t.h.koornwinder/

Received April 29, 2015, in final form September 14, 2015; Published online
September 20, 2015

For each of the eight $n$-th derivative parameter changing formulas for Gauss
hypergeometric functions a corresponding fractional integration formula is
given. For both types of formulas the differential or integral operator is
intertwining between two actions of the hypergeometric differential operator
(for two sets of parameters): a so-called transmutation property. This leads
to eight fractional integration formulas and four generalized Stieltjes
transform formulas for each of the six different explicit solutions of the
hypergeometric differential equation, by letting the transforms act on the
solutions. By specialization two Euler type integral representations for each
of the six solutions are obtained.

Gauss hypergeometric function; Euler integral representation; fractional
integral transform; Stieltjes transform; transmutation formula

33C05; 44A15; 44A20; 26A33

##  1 Introduction

This paper has two sources of inspiration. The first aim was to give a
complete list of the fractional integration formulas corresponding to the
eight parameter changing $n$-th derivative formulas for Gauss hypergeometric
functions given in [7, 2.8(20)–(27)] and [18, § 15.5]. The first fractional
generalization of one of these differentiation formulas was given by Bateman
[4, p. 184]. Fractional generalizations of some further differentiation
formulas were given by Askey & Fitch [3, Section 2]. One still missing case
was partially handled by Camporesi [5, paragraph after (2.28)]. In this paper
the full list of the eight fractional integral transformation formulas will be
given.

Another observation leading to this paper was that Euler’s integral
representation for the Gauss hypergeometric function, when written as a
fractional integral

$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};x\right)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}x^{1-c}\int_{0}^{x}y^{b-1}(1-y)^{-a}(x-y)^{c-b-1}\,dy$
$\displaystyle\hphantom{\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};x\right)=}{}\
(0<x<1,\;{\rm Re}\,c>{\rm Re}\,b>0),$ (1.1)

should have a proof by using the hypergeometric differential equation [1,
(2.3.5)]

$\displaystyle
L_{a,b,c;z}\left(\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)\right)=0,$
(1.2)

where

$\displaystyle
L_{a,b,c;z}\big{(}f(z)\big{)}=(L_{a,b,c}f)(z):=z(1-z)f^{\prime\prime}(z)+(c-(a+b+1)z)f^{\prime}(z)-abf(z),$
(1.3)

and that then essentially the same proof should also yield that

$\displaystyle
f(x):=|x|^{1-c}\int_{m}^{M}|y|^{b-1}|1-y|^{-a}|x-y|^{c-b-1}\,dy$ (1.4)

is a solution of the hypergeometric differential equation if $m$, $M$ and $x$
are as follows:

$(m,M)$ $(-\infty,x)$ $(-\infty,0)$ $(x,0)$ $(0,x)$ $(0,1)$ $(0,1)$ $(x,1)$
$(1,x)$ $(1,\infty)$ $(x,\infty)$ $x\in$ $(-\infty,0)$ $(0,\infty)$
$(-\infty,0)$ $(0,1)$ $(-\infty,0)$ $(1,\infty)$ $(0,1)$ $(1,\infty)$
$(-\infty,1)$ $(1,\infty)$

This means that $m$ and $M$ in (1.4) are two consecutive points of singularity
of the integrand. Indeed, we will see that in all listed cases the right-hand
side of (1.4) equals a constant multiple of one of the six explicit solutions
$w_{j}$ [18, § 15.10(ii)] of the hypergeometric differential equation. In
fact, as was Sergei Sitnik kindly commenting to me following an earlier
version of this paper, the above observations (in the fractional integral case
where $m$ or $M$ equals $x$) were already made in great detail by Letnikov
[14] in 1874 in a paper which seems to have been unobserved outside Russia.

The Euler type integral representations of fractional integral type are
specializations of parameter changing fractional integral transforms acting on
some $w_{j}$. For instance, (1.1) is a special case of Bateman’s fractional
integral formula [4, p. 184]:

$\displaystyle\int_{0}^{x}y^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c)}{\Gamma(c+\mu)}x^{c+\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c+\mu};x\right)$
$\displaystyle\hphantom{\int_{0}^{x}y^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=}{}(0<x<1,\;{\rm
Re}\,c>0,\;{\rm Re}\,\mu>0).$ (1.5)

It will turn out that (1.5), and all other fractional integral formulas for
hypergeometric functions to be considered, admit a proof by using the
hypergeometric differential equation. Here $L_{a,b,c}$, given by (1.3), occurs
in so-called transmutation formulas, for instance in connection with (1.5):

$\displaystyle
L_{a,b,c+\mu;x}\left(x^{1-c-\mu}\\!\int_{0}^{x}y^{c-1}f(y)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy\right)=x^{1-c-\mu}\\!\int_{0}^{x}y^{c-1}(L_{a,b,c}f)(y)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy$
$\displaystyle\hskip 142.26378pt(f\in C^{2}([0,1)),\;x\in(0,1),\;{\rm
Re}\,c>0,\;{\rm Re}\,\mu>0).$ (1.6)

Transmutation is a term which occurs in many meanings in science, and even can
have various meanings in mathematics, but in the sense used here it was first
considered in great detail by Lions [15], namely as an operator $A$, often an
integral operator, intertwining between two differential operators $L_{1}$ and
$L_{2}$:

$\displaystyle L_{1}A=AL_{2}.$

In most examples, but not here, $L_{1}=d^{2}/dx^{2}$. Then a typical example
for $L_{2}$ would be the Bessel type differential operator
$L_{2}=d^{2}/dx^{2}+ax^{-1}d/dx$. Lions [15] built on earlier work by J.
Delsarte (1938) and Levitan (1951). Many papers and books on transmutation
have appeared since then. See the surveys [21, 22] by Sitnik and references
given there.

The cases of (1.4) where $m$ and $M$ are not equal to $x$ are variants of the
generalized Stieltjes transform, introduced by Widder [26, Section 8], and
further developed by many authors, see for instance references in [10, 11]. We
consider the generalized Stieltjes transform as a transform sending $f$ to $g$
of the form

$\displaystyle g(x)=\int_{m}^{M}f(y)|y-x|^{\mu-1}\,dy,$ (1.7)

where $(m,M)$ is $(-\infty,0)$ or $(0,1)$ or $(1,\infty)$, where $x$ is on
$\mathbb{R}$ outside the integration interval, and where the function
$y\mapsto|y|^{\mu-1}f(y)$ is $L^{1}$ on the integration interval. The Euler
type integral representations of this form are special cases of generalized
Stieltjes transforms which send a solution $w_{i}$ of the hypergeometric
differential equation to a solution $w_{j}$ and change the parameters.

Some of these formulas can be found in literature, notably in the Bateman
project [8]. There [8, 14.4(9)] is essentially the case $(m,M)=(-\infty,0)$ of
(1.4), while [8, 20.2(10)] is a generalized Stieltjes transform sending a
${}_{2}F_{1}$ to a ${}_{2}F_{1}$. A formula by Karp & Sitnik [12, Lemma 2] is
essentially a generalized Stieltjes transform sending a ${}_{2}F_{1}$ to a
${}_{3}F_{2}$, which can be specialized to a transform sending ${}_{2}F_{1}$
to ${}_{2}F_{1}$. We will give a long list of generalized Stieltjes transforms
sending some $w_{i}$ to some $w_{j}$, including the two from literature just
mentioned. The formulas in this list are essentially equivalent: they can all
be derived from each other.

The case $\mu=0$ of (1.7) is essentially the classical Stieltjes transform. It
will not change the parameters of the hypergeometric solutions. A well-known
example of this case is [25, (4.61.4)], which sends Jacobi polynomials to
Jacobi functions of the second kind.

The idea that formulas for hypergeometric functions can be proved by using the
hypergeometric differential equation goes back to Riemann. See Andrews, Askey
& Roy [1, Sections 2.3 and 3.9] how this method can be used for a proof of
Pfaff’s and Euler’s transformation formulas and of quadratic transformation
formulas. Such methods are recently also used by Paris and coauthors [13, 24].
They refer to an earlier proof in this spirit by Rainville [19, p. 126] of a
quadratic transformation formula involving a ${}_{1}F_{1}$ and a
${}_{0}F_{1}$. In this paper the method will be applied to integral formulas,
but with a different focus.

Our message is that many formulas for hypergeometric functions have a
companion formula involving the hypergeometric differential equation, which is
more universal because it will imply or suggest many formulas involving the
various solutions $w_{j}$ of the hypergeometric differential equation. A final
rigorous proof of these formulas may not use the universal formula (as will be
often the case in the present paper), but the universal formula is helpful for
arriving at these formulas and for organizing them.

Quite probably the ideas of this paper will also work in other situations, for
instance for Appell hypergeometric series.

The contents of this paper are as follows. After some preliminaries for
hypergeometric functions in Section 2, we illustrate in Section 3 the main
ideas of the paper for the special case of the Bateman integral (1.5). This
takes quite a few pages, but it is less technical than the rest of the paper.
In Section 4 we state and prove the eight fractional integral transformations
corresponding to the eight $n$-th derivative formulas for hypergeometric
functions. We give also eight corresponding transmutation formulas. In Section
5 we discuss the 48 fractional integration formulas for the six solutions
$w_{j}$, which can be obtained by rewriting the formulas in Section 4. Not all
formulas will be given explicitly. In Section 6 we give 24 generalized
Stieltjes transforms sending some $w_{i}$ to some $w_{j}$. In Section 7 we
give for each of the six solutions $w_{j}$ two Euler type integral
representations. They can all be obtained by specialization of formulas in
Sections 5 and 6. Finally, Section 8 discusses the connection between
generalized Stieltjes transforms of different order by fractional integration,
and how this leads to connections between formulas in Sections 5 and 6.

##  2 Preliminaries about hypergeometric functions

The Gauss hypergeometric function [7, Chapter 2], [1, Chapter 2], [18, Chapter
15] is defined by its power series

$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=F(a,b;c;z):=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}k!}z^{k}\qquad(|z|<1).$
(2.1)

Here the (complex) parameters $a,b,c$ are taken generically. In particular the
series should be well-defined ($c\notin\mathbb{Z}_{\leq 0}$), while possibly
nicer results in case of terminating series ($a$ or $b\in\mathbb{Z}_{\leq 0}$)
are not paid attention to. The function (2.1) has a one-valued analytic
continuation to $\mathbb{C}\backslash[1,\infty)$. It satisfies the
hypergeometric differential equation (1.2) and it is uniquely determined as
the function regular near 0 and equal to 1 at 0 which is annihilated by
$L_{a,b,c}$.

Important transformation formulas, due to Pfaff and Euler, respectively, are
[1, (2.2.6), (2.2.7)], [18, (15.8.1)]:

$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=(1-z)^{-a}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,c-b}{c};\frac{z}{z-1}\right)\qquad$
$\displaystyle(z\notin[1,\infty)),$ (2.2)
$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=(1-z)^{c-a-b}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{c-a,c-b}{c};z\right)\qquad$
$\displaystyle(z\notin[1,\infty)).$ (2.3)

Often, a specific formula for $F(a,b;c;\,.\,)$ trivially implies another one
by the symmetry in $a$ and $b$. For instance, when we will refer to (2.2), we
may also mean a similar identity with the right-hand side given by
$(1-z)^{-b}F(c-a,b;c;z/(z-1))$. An elementary special case of the
hypergeometric function is

$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{b};z\right)=(1-z)^{-a}\qquad(z\notin[1,\infty)).$
(2.4)

In the case of generic parameters there are essentially six different explicit
solutions of the Gauss differential equation $L_{a,b,c}f=0$ [7, Section 2.9],
[18, § 15.10(ii)]. These are one-valued analytic functions on the complex
plane with a suitable real interval as cut:

$\displaystyle
w_{1}(z;a,b,c):=\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)\qquad$
$\displaystyle(z\notin[1,\infty)),$ (2.5) $\displaystyle
w_{2}(z;a,b,c):=z^{1-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-c+1,b-c+1}{2-c};z\right)\quad$
$\displaystyle(z\notin(-\infty,0]\cup[1,\infty)),$ (2.6) $\displaystyle
w_{3}(z;a,b,c):=z^{-a}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,a-c+1}{a-b+1};z^{-1}\right)\qquad$
$\displaystyle(z\notin(-\infty,1]),$ (2.7) $\displaystyle
w_{4}(z;a,b,c):=z^{-b}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{b,b-c+1}{b-a+1};z^{-1}\right)\qquad$
$\displaystyle(z\notin(-\infty,1]),$ (2.8) $\displaystyle
w_{5}(z;a,b,c):=\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{a+b-c+1};1-z\right)\qquad$
$\displaystyle(z\notin(-\infty,0]),$ (2.9) $\displaystyle
w_{6}(z;a,b,c):=(1-z)^{c-a-b}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{c-a,c-b}{c-a-b+1};1-z\right)\qquad$
$\displaystyle(z\notin(-\infty,0]\cup[1,\infty)).$ (2.10)

Here we had to exclude not just the branch cuts of the ${}_{2}F_{1}$’s, but
also those of the power factors (we assume principal values for the complex
powers). In the case of $w_{2}$, $w_{3}$, $w_{4}$, $w_{6}$ we might have
chosen the branch cuts due to the power factors differently. For instance, a
companion of $w_{2}$ would be

$\displaystyle\widetilde{w}_{2}(z;a,b,c):=(-z)^{1-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-c+1,b-c+1}{2-c};z\right)\qquad(z\notin[0,\infty)).$

We will also consider the six solutions on subintervals of the real axis
including the branch cuts of the power factors as below (here we abuse
notation by not changing it compared to above)

$\displaystyle
w_{1}(x;a,b,c)=\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};x\right)$
$\displaystyle(x\in(-\infty,1)),$ (2.11) $\displaystyle
w_{2}(x;a,b,c)=|x|^{1-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-c+1,b-c+1}{2-c};x\right)$
$\displaystyle(x\in(-\infty,0)\cup(0,1)),$ (2.12) $\displaystyle
w_{3}(x;a,b,c)=|x|^{-a}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,a-c+1}{a-b+1};x^{-1}\right)$
$\displaystyle(x\in(-\infty,0)\cup(1,\infty)),$ (2.13) $\displaystyle
w_{4}(x;a,b,c)=|x|^{-b}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{b,b-c+1}{b-a+1};x^{-1}\right)$
$\displaystyle(x\in(-\infty,0)\cup(1,\infty)),$ (2.14) $\displaystyle
w_{5}(x;a,b,c)=\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{a+b-c+1};1-x\right)$
$\displaystyle(x\in(0,\infty)),$ (2.15) $\displaystyle
w_{6}(x;a,b,c)=|1-x|^{c-a-b}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{c-a,c-b}{c-a-b+1};1-x\right)\qquad$
$\displaystyle(x\in(0,1)\cup(1,\infty)).$ (2.16)

##  3 The main ideas illustrated in a special case

###  3.1 Transmutation property of a differentiation operator

In [7, 2.8(20)–(27)] or [18, § 15.5] there is a list of eight parameter
changing $n$-th derivative formulas for Gauss hypergeometric functions. One of
these is [18, (15.5.4)]:

$\displaystyle\left(\frac{d}{dz}\right)^{n}\left(z^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)\right)=\frac{\Gamma(c)}{\Gamma(c-n)}\,z^{c-n-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c-n};z\right).$
(3.1)

Formula (3.1) can be proved immediately by power series expansion (2.1). Note
also that the $n$-th derivative case follows by iteration of the case $n=1$.
The case $n=1$ can be rewritten as

$\displaystyle
D_{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};\,.\,\right)=(c-1)\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c-1};\,.\,\right),$
(3.2)

where

$\displaystyle(D_{a}f)(z):=zf^{\prime}(z)+af(z)=z^{-a+1}\frac{d}{dz}\big{(}z^{a}f(z)\big{)}.$
(3.3)

Clearly, if $f$ is analytic at 0 then $D_{a}f$ is analytic at 0 and
$(D_{a}f)(0)=af(0)$.

Straightforward computation followed by iteration gives the following
transmutation property:

$\displaystyle L_{a,b,c-1}D_{c-1}=D_{c-1}L_{a,b,c},$ (3.4) $\displaystyle
L_{a,b,c-n}D_{c-n}\cdots D_{c-2}D_{c-1}=D_{c-n}\cdots
D_{c-2}D_{c-1}L_{a,b,c}.$ (3.5)

Formula (3.2) is also a consequence of (3.4). Indeed, by (3.4) $L_{a,b,c-1}$
annihilates the left-hand side of (3.2). Since this left-hand side is regular
at 0, it must be a constant times $F(a,b;c-1;\,.\,)$ with the constant
obtained by evaluating both sides at 0. Similarly, (3.1) is a consequence of
(3.5).

Because of the above argument, it is natural to consider
$D_{c-1}w_{j}(\,.\,;a,b,c)$ not just for $j=1$ but also for the other explicit
solutions of the hypergeometric differential equation ($j=2,\ldots,6$). It
will turn out that for all $j$ we get some constant factor times
$w_{j}(\,.\,;a,b,c-1)$. For instance, with $w_{2}$ given by (2.6),

$\displaystyle
D_{c-1}w_{2}(\,.\,;a,b,c)=\frac{(a-c+1)(b-c+1)}{2-c}w_{2}(\,.\,;a,b,c-1).$
(3.6)

This is equivalent to the prototypical differentiation formula for the
hypergeometric function [4, (15.5.4)]:

$\displaystyle\frac{d}{dz}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=\frac{ab}{c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a+1,b+1}{c+1};z\right).$
(3.7)

Corresponding to each of the eight $n$-th derivative formulas [18,
(15.5.2)–(15.5.9)] one can write down a transmutation formula like (3.5), and
corresponding to each of these transmutation formulas one can write down an
$n$-th derivative formula for each of the six solutions $w_{j}$. These will
not be included in the present paper. The transmutation formulas for the first
derivative operators were earlier given by Dereziński [6, end of Section 3.4].
He calls them commutation relations and he relates them to to the Lie algebra
$\mathfrak{so}(6)$ (or equivalently $\mathfrak{sl}(4)$). This way to associate
a Lie algebra (the so-called dynamical symmetry algebra) with a hypergeometric
differential equation was first described by Miller [17]. This was later
understood in the framework of $\cal A$-hypergeometric systems, see Saito
[20].

######  Remark 3.1.

The transformation formulas (2.2) and (2.3) also have a more universal
background because they can be understood from transformation properties of
the differential operator $L_{a,b,c}$ (see also [1, (2.3.10B) and (2.3.10F)]):

$\displaystyle
L_{a,b,c;z}\left((1-z)^{-a}f\left(\frac{z}{z-1}\right)\right)=-(1-z)^{-a-1}\big{(}L_{a,c-b,c}f\big{)}\left(\frac{z}{z-1}\right),$
(3.8) $\displaystyle
L_{a,b,c;z}\big{(}(1-z)^{c-a-b}f(z)\big{)}=(1-z)^{c-a-b}(L_{c-a,c-b,c}f)(z).$
(3.9)

This suggests identities involving $w_{i}(z;a,b,c)$,
$(1-z)^{-a}w_{j}(\frac{z}{z-1};a,c-b,c)$, $(1-z)^{-b}w_{k}(\frac{z}{z-1};$
$c-a,b,c)$ and $(1-z)^{c-a-b}w_{l}(z;c-a,c-b,c)$. These can indeed be given,
but sometimes one needs another branch cut for the power factor than chosen in
(2.5)–(2.10).

###  3.2 Transmutation property of a fractional integral operator

Both Riemann–Liouville and Weyl fractional integrals will occur in our
formulas. See, for instance, [8, pp. 181–183] or [16].

Formula (3.1) can equivalently be written as a repeated integral, which can be
condensed to a single integral [1, p. 111]. For this purpose, take $0<x<1$ and
assume ${\rm Re}\,c>n$. We obtain

$\displaystyle
x^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};x\right)=\frac{\Gamma(c)}{\Gamma(c-n)}\int_{0}^{x}y^{c-n-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c-n};y\right)\frac{(x-y)^{n-1}}{(n-1)!}\,dy.$
(3.10)

Note that there are no other terms because, by our assumption, $y^{c-m}$
vanishes at $y=0$ for $m=1,\ldots,n$. Formula (3.10) has a fractional
extension for $n$ complex with ${\rm Re}\,c>{\rm Re}\,n>0$. In rewritten form
this is Bateman’s fractional integral formula (1.5), which is often written in
the form

$\displaystyle\int_{0}^{1}t^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};tz\right)\frac{(1-t)^{\mu-1}}{\Gamma(\mu)}\,dt=\frac{\Gamma(c)}{\Gamma(c+\mu)}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c+\mu};z\right)$
$\displaystyle\hphantom{\int_{0}^{1}t^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};tz\right)\frac{(1-t)^{\mu-1}}{\Gamma(\mu)}\,dt=}{}\
(z\in\mathbb{C}\backslash[1,\infty),\;{\rm Re}\,c>0,\;{\rm Re}\,\mu>0).$
(3.11)

Then (1.5) follows from (3.11) by specializing $z$ to $x\in(0,1)$ and
substituting $t=y/x$. By analytic continuation it is sufficient to prove
(3.11) for $|z|<1$. There it follows by power series expansion (2.1). For
(3.11) see also [3, (2.4)], [2, (3.5)] and [1, Theorem 2.2.4, (2.9.6)].

Just as we proved (3.2) by (3.4), we can prove (1.5) from the following
identity (derived by straightforward computation):

$\displaystyle
y^{1-c}L_{1-a,1-b,2-c;y}\big{(}y^{c-1}(x-y)^{\mu-1}\big{)}=x^{c+\mu-1}L_{a,b,c+\mu;x}\big{(}x^{1-c-\mu}(x-y)^{\mu-1}\big{)}.$
(3.12)

We also need (straightforward by integration by parts):

######  Lemma 3.2.

Let $f,g\in C^{2}((m,M))$. Then, for $x\in(m,M)$,

$\displaystyle(L_{a,b,c}f)(x)g(x)-f(x)(L_{1-a,1-b,2-c}g)(x)$
$\displaystyle\qquad{}=\frac{d}{dx}\big{(}x(1-x)\big{(}f^{\prime}(x)g(x)-f(x)g^{\prime}(x)\big{)}+\big{(}c-1+(1-a-b)x\big{)}f(x)g(x)\big{)}.$

Furthermore, let $x(1-x)f^{\prime\prime}(x)g(x)$,
$x(1-x)f(x)g^{\prime\prime}(x)$, $(1+|x|)f^{\prime}(x)g(x)$,
$(1+|x|)f(x)g^{\prime}(x)$, $f(x)g(x)$, as functions of $x$, be in
$L^{1}((m,M))$, and assume that

$\displaystyle\left.\begin{matrix}x(1-x)f^{\prime}(x)g(x)\\\\[3.0pt]
x(1-x)f(x)g^{\prime}(x)\\\\[3.0pt] (1+|x|)f(x)g(x)\end{matrix}\right\\}\to
0\quad\mbox{\rm as $x\downarrow m$ or $x\uparrow M$.}$

Then

$\displaystyle\int_{m}^{M}(L_{a,b,c}f)(x)g(x)\,dx=\int_{m}^{M}f(x)(L_{1-a,1-b,2-c}g)(x)\,dx.$
(3.13)

In particular, $L_{1-a,1-b,2-c}$ is the formal adjoint of $L_{a,b,c}$.

###### Proof of (1.5) by (3.12).

First assume $0<x<1$, ${\rm Re}\,c>1$, ${\rm Re}\,\mu>2$, We have

$\displaystyle L_{a,b,c+\mu;x}\big{(}x^{1-c-\mu}\times\mbox{left-hand side of
\eqref{1}}\big{)}$
$\displaystyle\qquad{}=\frac{1}{\Gamma(\mu)}\int_{0}^{x}y^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)L_{a,b,c+\mu;x}\big{(}x^{1-c-\mu}(x-y)^{\mu-1}\big{)}\,dy$
$\displaystyle\qquad{}=\frac{x^{1-c-\mu}}{\Gamma(\mu)}\int_{0}^{x}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)L_{1-a,1-b,2-c;y}\big{(}y^{c-1}(x-y)^{\mu-1}\big{)}\,dy$
$\displaystyle\qquad{}=\frac{x^{1-c-\mu}}{\Gamma(\mu)}\int_{0}^{x}L_{a,b,c;y}\left(\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\right)y^{c-1}(x-y)^{\mu-1}\,dy=0.$

For the first equality sign we twice used a generalized Leibniz rule [9,
(1.1)] (allowed because the integral in the second line converges absolutely)
together with the vanishing of the integrand of the left-hand side of (1.5)
for $y=x$ (and similarly for the derivative with respect to $x$ of that
integrand)111The involved differentiation under the integral sign got some
fame as “Feynman’s trick”, see
https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign#Popular_culture..
For the second equality apply (3.12) and for the third equality (3.13) (the
conditions are satisfied). Because

$\displaystyle x^{1-c-\mu}\times\mbox{left-hand side of
\eqref{1}}=\int_{0}^{1}t^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};tx\right)\frac{(1-t)^{\mu-1}}{\Gamma(\mu)}\,dt,$

this is analytic in $x$ at $x=0$, and for $x=0$ it takes the value

$\displaystyle\int_{0}^{1}t^{c-1}\frac{(1-t)^{\mu-1}}{\Gamma(\mu)}\,dt=\frac{\Gamma(c)}{\Gamma(c+\mu)}.$

Thus, by the characterization of the hypergeometric function as solution of
the hypergeometric differential equation, we have proved (3.12) for ${\rm
Re}\,\mu>2$, ${\rm Re}\,c>1$. These conditions can be relaxed by analytic
continuation. ∎

If we replace in the above proof the ${}_{2}F_{1}$ by a suitable function $f$
then we obtain the transmutation property (1.6).

######  Remark 3.3.

For ${\rm Re}\,\mu>2$ (1.6) is equivalent to (3.12). For $\mu=3,4,\ldots$
(1.6) is also equivalent to (3.5). Furthermore, it is sufficient to prove
(3.12) for $\mu=3,4,\ldots$ because, after division of both sides by
$(x-y)^{\mu-1}$, the two sides depend polynomially on $\mu$.

Formally, the proof of (1.6) can be extended to showing that

$\displaystyle
L_{a,b,c+\mu;x}\left(|x|^{1-c-\mu}\int_{m}^{M}|y|^{c-1}f(y)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy\right)$
$\displaystyle\qquad{}=|x|^{1-c-\mu}\int_{m}^{M}|y|^{c-1}(L_{a,b,c}f)(y)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy.$
(3.14)

Here $m$ or $M$ may be equal to $x$, but not necessarily, and $y$ and $x-y$
should not change sign for $y\in[m,M)$. This becomes rigorous if $f$, $c$ and
$\mu$ are such that the first and third equality (suitably modified) in the
above Proof remain valid in the case of (3.14). In particular, $f$ should have
suitable vanishing properties at $m$ and $M$, even stronger if $m$ or $M$ are
not equal to $0$ or $x$.

If (3.14) holds and if $L_{a,b,c}f=0$ then

$\displaystyle L_{a,b,c+\mu}g=0,\qquad
g(x):=|x|^{1-c-\mu}\int_{m}^{M}|y|^{c-1}f(y)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy.$
(3.15)

In this paper many explicit cases of (3.15) will be given, where $f=w_{i}$ for
some $i$ and $g={\rm const}\cdot w_{j}$ for some $j$, with $j=i$ if $m$ or $M$
equals $x$. As an example it can be derived that

$\displaystyle\int_{-\infty}^{x}(-y)^{c-1}w_{2}(y;a,b,c)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-c-\mu+1)}{\Gamma(a-c+1)}\frac{\Gamma(b-c-\mu+1)}{\Gamma(b-c+1)}\frac{\Gamma(2-c)}{\Gamma(2-c-\mu)}$
$\displaystyle\hphantom{\int_{-\infty}^{x}(-y)^{c-1}w_{2}(y;a,b,c)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=}{}\times(-x)^{c+\mu-1}w_{2}(x;a,b,c+\mu)$
(3.16) $\displaystyle\hskip 142.26378pt{}(x<0,\;{\rm Re}\,(a-c-\mu+1),{\rm
Re}\,(b-c-\mu+1),{\rm Re}\,\mu>0).$

This is the fractional generalization of the iteration of (3.6). It can be
equivalently written as

$\displaystyle\int_{-\infty}^{x}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-\mu)}{\Gamma(a)}\frac{\Gamma(b-\mu)}{\Gamma(b)}\frac{\Gamma(c)}{\Gamma(c-\mu)}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b-\mu}{c-\mu};x\right)$
$\displaystyle\hphantom{\int_{-\infty}^{x}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=}{}\
(x<0,\;{\rm Re}\,a,\;{\rm Re}\,b>{\rm Re}\,\mu>0).$ (3.17)

This is the fractional generalization of the iteration of (3.7). A different
proof will be given in the next section.

Another explicit case of (3.15) which we will meet is

$\displaystyle\int_{1}^{\infty}y^{c-1}w_{6}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-c-\mu+1)\Gamma(b-c-\mu+1)\Gamma(c-a-b+1)}{\Gamma(2-c-\mu)\Gamma(1-\mu)}x^{c+\mu-1}w_{2}(x;a,b,c+\mu)$
$\displaystyle\qquad\quad(0<x<1,\;{\rm Re}\,(a-c-\mu+1),\;{\rm
Re}\,(b-c-\mu+1),\;{\rm Re}\,(c-a-b+1)>0).$ (3.18)

The left-hand side is no longer of fractional integral type, but it is a
generalized Stieltjes transform, to which we will return in a moment.

###  3.3 Euler type integral representations

When we replace $c$, $\mu$ by $b$, $c-b$ in (1.5) or (3.11) and use (2.4) then
we obtain Euler’s integral representation as fractional integral (1.1), or in
its most used form

$\displaystyle\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}\,dt$
$\displaystyle\hphantom{\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};z\right)=}{}\
(z\in\mathbb{C}\backslash[1,\infty),\;{\rm Re}\,c>{\rm Re}\,b>0).$ (3.19)

The right-hand side of (1.1) is annihilated by $L_{a,b,c;x}$. This is a
consequence of the transmutation property (1.6), by which $L_{a,b,c;x}$ acting
on the right-hand side of (1.1) is equal to

$\displaystyle{\rm const}\cdot
x^{1-c}\int_{0}^{x}y^{b-1}L_{a,b,b;y}\big{(}(1-y)^{-a}\big{)}(x-y)^{c-b-1}\,dy=0$

because $(1-y)^{-a}$ is annihilated by $L_{a,b,b;y}$. This last fact is also
apparent from

$\displaystyle
L_{a,b,b;y}=\left(y\frac{d}{dy}+b\right)\circ\left((1-y)\frac{d}{dy}-a\right).$

The more general transmutation property (3.14), considered with $c,\mu$
replaced by $b$, $c-b$, suggests that each $w_{j}$ has an Euler type integral
representation (1.4). This is indeed the case, as we already briefly indicated
after (1.4).

We can write (3.19) also equivalently as a generalized Stieltjes transform

$\displaystyle
w_{3}(x;a,b,c)=\frac{\Gamma(a-b+1)}{\Gamma(a-c+1)\Gamma(c-b)}\int_{0}^{1}y^{a-c}(1-y)^{c-b-1}(x-y)^{-a}\,dy$
$\displaystyle\hphantom{w_{3}(x;a,b,c)=}{}\ (x>1,\;{\rm Re}\,(a+1)>{\rm
Re}\,c>{\rm Re}\,b).$ (3.20)

It will turn out that, more generally and analogous to (1.4),

$\displaystyle g(x)=\int_{m}^{M}|y|^{a-c}|1-y|^{c-b-1}|x-y|^{-a}\,dy$ (3.21)

is a solution of the hypergeometric differential equation if $m$, $M$ and $x$
are as listed after (1.4).

Although most Euler type integral representations of fractional integral type
are equivalent to some integral representation of generalized Stieltjes
transform type by a change of integration variable, this is no longer true for
the integral transforms mapping $w_{i}$ to $w_{j}$ which specialize to an
Euler type integral representation.

###  3.4 Generalized Stieltjes transforms as transmutation operators

We already observed in Section 1 that variants of fractional integral
transforms and the Euler integral representation for hypergeometric functions
naturally lead to formulas involving a generalized Stieltjes transform. In the
definition by Widder [26, Section 8] the generalized Stieltjes transform sends
a suitable measure $\alpha$ or function $\phi$ (with $d\alpha(t)=\phi(t)\,dt$)
to a function $f$ analytic on $\mathbb{C}\backslash(-\infty,0]$:

$\displaystyle\int_{0}^{\infty}\frac{d\alpha(t)}{(z+t)^{\rho}}=f(z).$

The special case $\rho=1$ gives the classical Stieltjes transform. In order to
have analytic expressions similar to the ones in fractional integral
transforms, we will work with transforms (1.7).

Transforms of generalized Stieltjes type have transmutation properties. For
instance, from (3.14) we have, associated with (3.18), the intertwining
property

$\displaystyle
L_{a,b,c+\mu;x}\left(x^{1-c-\mu}\int_{1}^{\infty}y^{c-1}f(y)(y-x)^{\mu-1}\,dy\right)$
$\displaystyle\qquad{}=x^{1-c-\mu}\int_{1}^{\infty}y^{c-1}(L_{a,b,c}f)(y)(y-x)^{\mu-1}\,dy\qquad(0<x<1).$

Noteworthy is the case $\mu=0$. Then we have the same hypergeometric
differential operator on both sides. The Stieltjes transform will then map
solutions of the differential equation to other solutions.

Karp & Sitnik [12, Lemma 2] proved the following formula:

$\displaystyle\int_{0}^{1}t^{b-1}(1-t)^{d+e-b-c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{d-c,e-c}{d+e-b-c};1-t\right)(1-tz)^{-a}\,dt$
$\displaystyle\qquad{}=\frac{\Gamma(b)\Gamma(c)\Gamma(d+e-b-c)}{\Gamma(d)\Gamma(e)}\,\mbox{}_{3}F_{2}\\!\left(\genfrac{}{}{0.0pt}{}{a,b,c}{d,e};z\right)$
$\displaystyle\qquad\quad\ (z\in\mathbb{C}\backslash[1,\infty),\;{\rm
Re}\,b,{\rm Re}\,c,{\rm Re}\,(d+e-b-c)>0).$ (3.22)

If we replace $z$ by $z^{-1}$ then we recognize the formula as a generalized
Stieltjes transform sending a ${}_{2}F_{1}$ to a ${}_{3}F_{2}$ of general
parameters. (For more general formulas expressing ${}_{p}F_{p-1}$ functions as
generalized Stieltjes transforms see [11] and references given there.) If
moreover $d=a$ the transform in [19] sends a ${}_{2}F_{1}$ to a ${}_{2}F_{1}$:

$\displaystyle\int_{0}^{1}t^{b-1}(1-t)^{a+e-b-c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-c,e-c}{a+e-b-c};1-t\right)(z-t)^{-a}\,dt$
$\displaystyle\qquad{}=\frac{\Gamma(b)\Gamma(c)\Gamma(a+e-b-c)}{\Gamma(a)\Gamma(e)}z^{-a}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{b,c}{e};z^{-1}\right)$
$\displaystyle\qquad\quad\ (z\in\mathbb{C}\backslash(-\infty,1],\;{\rm
Re}\,b,{\rm Re}\,c,{\rm Re}\,(d+e-b-c)>0).$ (3.23)

All generalized Stieltjes transforms mapping a solution $w_{i}$ to a solution
$w_{j}$ we know can be obtained from (3.23) by change of parameters, change of
integration variable, and application of (2.2) and (2.3).

##  4 The eight fractional integral transformations
of the Gauss hypergeometric function

Each of the eight $n$-th derivative formulas in [18, § 15.5] has a fractional
generalization. Some of these are very well-known, but others were hardly
known until now. They fall apart into three families. Within a family the
formulas follow from each other by application of (2.2) or (2.3). We will use
shorthand names for the eight cases of which the meaning will be obvious. The
division of the cases into families and their correspondence with the $n$-th
derivative formulas is as follows:

family I I I II II III III III case $c+$ $a+$, $c+$ $a+$, $b+$, $c+$ $a-$ $a+$
$a-$, $b-$, $c-$ $a-$, $c-$ $c-$ [18, § 15.5] (4) (8) (9) (3) (5) (1) (7) (6)

###  4.1 Family I

#### Case $\boldsymbol{c+}$.

We already discussed Bateman’s fractional integral formula (1.5). From (3.11)
we get another variant of (1.5):

$\displaystyle\int_{x}^{0}(-y)^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(y-x)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c)}{\Gamma(c+\mu)}(-x)^{c+\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c+\mu};x\right)$
$\displaystyle\hphantom{\int_{x}^{0}(-y)^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(y-x)^{\mu-1}}{\Gamma(\mu)}\,dy=}\
(x<0,\;{\rm Re}\,c>0,\;{\rm Re}\,\mu>0),$

which we can write together with (1.5) in a unified way as:

$\displaystyle\int_{0<y/x<1}|y|^{c-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c)}{\Gamma(c+\mu)}|x|^{c+\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c+\mu};x\right)$
$\displaystyle\qquad(x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,c>0,\;{\rm
Re}\,\mu>0).$ (4.1)

#### Case $\boldsymbol{a+}$, $\boldsymbol{c+}$.

By using (2.2) in (4.1) we arrive at

$\displaystyle\int_{0<y/x<1}|y|^{c-1}(1-y)^{b-c-\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c)}{\Gamma(c+\mu)}|x|^{c+\mu-1}(1-x)^{b-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a+\mu,b}{c+\mu};x\right)$
$\displaystyle\qquad\quad\ (x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,c>0,\;{\rm
Re}\,\mu>0).$ (4.2)

#### Case $\boldsymbol{a+}$, $\boldsymbol{b+}$, $\boldsymbol{c+}$.

By using (2.3) in (4.1) we arrive at

$\displaystyle\int_{0<y/x<1}|y|^{c-1}(1-y)^{a+b-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c)}{\Gamma(c+\mu)}|x|^{c+\mu-1}(1-x)^{a+b-c+\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a+\mu,b+\mu}{c+\mu};x\right)$
$\displaystyle\qquad\quad(x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,c>0,\;{\rm
Re}\,\mu>0).$ (4.3)

See also Askey & Fitch [3, (2.11)].

###  4.2 Family II

#### Case $a-$.

Askey & Fitch [3, (2.10)] give

$\displaystyle\int_{0}^{1}t^{a-\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};zt\right)\frac{(1-t)^{\mu-1}}{\Gamma(\mu)}\,dt=\frac{\Gamma(a-\mu)}{\Gamma(a)}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b}{c};z\right)$
$\displaystyle\qquad(z\notin(1,\infty),\;{\rm Re}\,a>{\rm Re}\,\mu>0).$ (4.4)

The proof is as for (3.11). For $|z|<1$ the formula follows by power series
expansion (2.1), and next the general case follows by analytic continuation.
Formula (4.4) implies the following fractional integral formula:

$\displaystyle\int_{0<y/x<1}|y|^{a-\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-\mu)}{\Gamma(a)}|x|^{a-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b}{c};x\right)$
$\displaystyle\qquad(x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,a>{\rm Re}\,\mu>0).$
(4.5)

#### Case $\boldsymbol{a+}$.

By using (2.2) or (2.3) in (4.5) we arrive at

$\displaystyle\int_{0<y/x<1}|y|^{c-a-\mu-1}(1-y)^{a+b-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-a-\mu)}{\Gamma(c-a)}|x|^{c-a-1}(1-x)^{a+b-c+\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a+\mu,b}{c};x\right)$
$\displaystyle\qquad\quad\ (x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,(c-a)>{\rm
Re}\,\mu>0).$ (4.6)

###  4.3 Family III

#### Case $\boldsymbol{a-}$, $\boldsymbol{b-}$, $\boldsymbol{c-}$.

The following generalizes a formula of Camporesi [5, paragraph after (2.28)].

######  Proposition 4.1.

We have

$\displaystyle\int_{-\infty}^{x}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-\mu)}{\Gamma(a)}\frac{\Gamma(b-\mu)}{\Gamma(b)}\frac{\Gamma(c)}{\Gamma(c-\mu)}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b-\mu}{c-\mu};x\right)$
$\displaystyle\qquad(x<1,\;{\rm Re}\,a,{\rm Re}\,b>{\rm Re}\,\mu>0).$ (4.7)

###### Proof.

Formula (4.5) can be rewritten as:

$\displaystyle\int_{1<\frac{y}{x}}|y|^{-a}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y^{-1}\right)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-\mu)}{\Gamma(a)}|x|^{-a+\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b}{c};x^{-1}\right)$
$\displaystyle\qquad(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,a>{\rm
Re}\,\mu>0).$ (4.8)

Combination with [7, 2.10(2) and (5)] (or with [18, (15.8.2)]) yields (4.7)
for $x<0$. This, in its turn, implies by transformation of integration
variable and by analytic continuation that

$\displaystyle\int_{1}^{\infty}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};1+tz\right)\frac{(t-1)^{\mu-1}}{\Gamma(\mu)}\,dt=\frac{\Gamma(a-\mu)}{\Gamma(a)}\frac{\Gamma(b-\mu)}{\Gamma(b)}\frac{\Gamma(c)}{\Gamma(c-\mu)}$
$\displaystyle\qquad{}\times(-z)^{-\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b-\mu}{c-\mu};1+z\right)\qquad(z\notin[0,\infty),\;{\rm
Re}\,a>{\rm Re}\,\mu>0).$ (4.9)

From that we see that (4.7) holds for $x<1$. ∎

#### Case $\boldsymbol{a-}$, $\boldsymbol{c-}$.

By using (2.2) in (4.7) we arrive at

$\displaystyle\int_{x}^{1}(1-y)^{a-\mu-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(y-x)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-\mu)}{\Gamma(a)}\frac{\Gamma(c-b-\mu)}{\Gamma(c-b)}\frac{\Gamma(c)}{\Gamma(c-\mu)}$
$\displaystyle\qquad{}\times(1-x)^{a-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a-\mu,b}{c-\mu};x\right)\qquad(x<1,\;{\rm
Re}\,a,{\rm Re}\,(c-b)>{\rm Re}\,\mu>0).$ (4.10)

#### Case $\boldsymbol{c-}$.

By using (2.3) in (4.7) we arrive at

$\displaystyle\int_{-\infty}^{x}(1-y)^{a+b-c}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c};y\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c-a-\mu)}{\Gamma(c-a)}\frac{\Gamma(c-b-\mu)}{\Gamma(c-b)}\frac{\Gamma(c)}{\Gamma(c-\mu)}$
$\displaystyle\qquad{}\times(1-x)^{a+b-c+\mu}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{a,b}{c-\mu};x\right)\qquad\\!\\!(x<1,\;{\rm
Re}\,(c-a),{\rm Re}\,(c-b)>{\rm Re}\,\mu>0).\\!\\!\\!$ (4.11)

###  4.4 Transmutation formulas

Corresponding to case $c+$ above, we gave already (3.12), which gives rise to
the transmutation formula (1.6), and by which a proof of (4.1) can be given.
Analogues of (3.12) and (1.6) can be given for all cases above. These have the
general form

$\displaystyle
L_{a^{\prime},b^{\prime},c^{\prime};x}\left(\frac{w(y)}{v(x)}|x-y|^{\mu-1}\right)=L_{1-a,1-b,2-c;y}\left(\frac{w(y)w_{2}(y)}{v(x)v_{2}(x)}|x-y|^{\mu-1}\right)$
(4.12)

and

$\displaystyle
L_{a^{\prime},b^{\prime},c^{\prime};x}\left(\int_{I}f(y)\frac{w(y)}{v(x)}|x-y|^{\mu-1}\,dy\right)=\int_{I}(L_{a,b,c}f)(y)\frac{w(y)w_{2}(y)}{v(x)v_{2}(x)}|x-y|^{\mu-1}\,dy.$
(4.13)

The data to be used in these two formulas for the eight cases are specified in
the table below. In general, $w(y)$ is a product of powers of $|y|$ and $1-y$,
and $w_{2}(y)$ is equal to $|y|$, $1-y$ or $1$. Similarly for $v(x)$ and
$v_{2}(x)$, respectively. The variable $y$ in (4.12) ranges over an open
interval $I$ which is also the integration interval in (4.13). The interval
$I$ has endpoint $x$ at one side and endpoint $x_{0}=0,1$ or $-\infty$ at the
other side. The variable $x$ in (4.12) and (4.13) ranges over some open subset
$J$ of $\mathbb{R}$. The function $f$ in (4.13) should be in $C^{2}(J)$ and
should moreover satisfy certain growth conditions at $x_{0}$, to be specified
in a moment. The parameter $\mu$ can be arbitrarily complex in (4.12) but
should satisfy ${\rm Re}\,\mu>2$ in (4.13).

case $a^{\prime}$ $b^{\prime}$ $c^{\prime}$ $x_{0}$ $w(y)/v(x)$
$w_{2}(y)/v_{2}(x)$ $J$ $c+$ $a$ $b$ $c+\mu$ $0$ $|y|^{c-1}/|x|^{c+\mu-1}$ $1$
$(-\infty,0)\cup(0,1)$ $a+$, $c+$ $a+\mu$ $b$ $c+\mu$ $0$
$\displaystyle\frac{|y|^{c-1}(1-y)^{b-c-\mu}}{|x|^{c+\mu-1}(1-x)^{b-c}}$
$\displaystyle\frac{1-y}{1-x}$ $(-\infty,0)\cup(0,1)$ $a+$, $b+$, $c+$ $a+\mu$
$b+\mu$ $c+\mu$ $0$
$\displaystyle\frac{|y|^{c-1}(1-y)^{a+b-c}}{|x|^{c+\mu-1}(1-x)^{b-c}}$ $1$
$(-\infty,0)\cup(0,1)$ $a-$ $a-\mu$ $b$ $c$ $0$ $|y|^{a-\mu-1}/|x|^{a-1}$
$|y|/|x|$ $(-\infty,0)\cup(0,1)$ $a+$ $a+\mu$ $b$ $c$ $0$
$\displaystyle\frac{|y|^{c-a-\mu-1}(1-y)^{a+b-c}}{|x|^{c-a-1}(1-x)^{a+b-c+\mu}}$
$|y|/|x|$ $(-\infty,0)\cup(0,1)$ $a-$, $b-$, $c-$ $a-\mu$ $b-\mu$ $c-\mu$
$-\infty$ $1$ $1$ $(-\infty,1)$ $a-$, $c-$ $a-\mu$ $b$ $c-\mu$ $1$
$\displaystyle\frac{(1-y)^{a-\mu-1}}{(1-x)^{a-1}}$
$\displaystyle\frac{1-y}{1-x}$ $(-\infty,1)$ $c-$ $a$ $b$ $c-\mu$ $-\infty$
$\displaystyle\frac{(1-y)^{a+b-c}}{(1-x)^{a+b-c+\mu}}$ $1$ $(-\infty,1)$

Verification of (4.12) in the eight cases is by straightforward computation,
possibly using computer algebra.

In order to find the growth conditions in (4.13) for $f$ at $x_{0}$ we recall
that (4.13) is obtained from the string of equalities

$\displaystyle
L_{a^{\prime},b^{\prime},c^{\prime};x}\left(\int_{I}f(y)\frac{w(y)}{v(x)}|x-y|^{\mu-1}\,dy\right)=\int_{I}f(y)L_{a^{\prime},b^{\prime},c^{\prime};x}\left(\frac{w(y)}{v(x)}|x-y|^{\mu-1}\right)dy$
$\displaystyle\qquad{}=\int_{I}f(y)L_{1-a,1-b,2-c;y}\left(\frac{w(y)w_{2}(y)}{v(x)v_{2}(x)}|x-y|^{\mu-1}\right)dy$
$\displaystyle\qquad{}=\int_{I}(L_{a,b,c}f)(y)\frac{w(y)w_{2}(y)}{v(x)v_{2}(x)}|x-y|^{\mu-1}\,dy.$

In the various steps we have to consider the singularities for $y$ at $x$ and
at $x_{0}$. The first singularity is already taken into account by the
condition ${\rm Re}\,\mu>2$. As for the singularity at $x_{0}$ the first and
the third equality need extra assumptions (which will also imply that the four
parts of the string are well defined). For the first equality we need

$\displaystyle f(y)w(y)(1+|y|)^{{\rm Re}\,\mu-1}\ \mbox{as a function of $y$
is $L^{1}$ at $y=x_{0}$.}$

For the third equality the needed assumptions follow from Lemma 3.2:

$\displaystyle\left.\begin{array}[]{@{}l}f^{\prime\prime}(y)y(1-y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu-1}\\\\[3.0pt] f^{\prime}(y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu}\\\\[3.0pt]
f(y)y(1-y)\dfrac{d^{2}}{dy^{2}}\big{(}w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu-1}\big{)}\\\\[3.0pt]
f(y)\dfrac{d}{dy}\big{(}w(y)w_{2}(y)(1+|y|)^{{\rm Re}\,\mu}\big{)}\\\\[3.0pt]
f(y)w(y)w_{2}(y)(1+|y|)^{{\rm Re}\,\mu-1}\end{array}\right\\}\;\mbox{as
functions of $y$ are $L^{1}$ at $y=x_{0}$,}$

and

$\displaystyle\left.\begin{array}[]{@{}l}f^{\prime}(y)y(1-y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu-1}\\\\[3.0pt] f(y)y(1-y)\dfrac{d}{dy}\big{(}w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu-1}\big{)}\\\\[3.0pt] f(y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu}\end{array}\right\\}\to 0\;\mbox{as $y\to x_{0}$.}$

Thus, for given $f$, the identity (4.13) is settled under certain constraints
for $a$, $b$, $c$ and $\mu$. But then the identity will be valid under relaxed
constraints on $a$, $b$, $c$ and $\mu$ such that both sides of (4.13) are
analytic in these four parameters, i.e., if ${\rm Re}\,\mu>2$ and

$\displaystyle\left.\begin{array}[]{@{}l}f^{\prime\prime}(y)y(1-y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu-1}\\\\[3.0pt] f^{\prime}(y)w(y)w_{2}(y)(1+|y|)^{{\rm
Re}\,\mu}\\\\[3.0pt] f(y)w(y)(1+|y|)^{{\rm
Re}\,\mu-1}\end{array}\right\\}\;\mbox{as functions of $y$ are $L^{1}$ at
$y=x_{0}$.}$

##  5 Fractional integral transformations for the six solutions
of the hypergeometric differential equation

Formula (4.1) is a fractional integral transformation of type $c+$ for the
solution $w_{1}$ of the hypergeometric differential equation. It turns out
that for all six solutions $w_{i}$ there is such a transformation of the form
$(\,.\,)^{c-1}w_{i}(\,.\,;a,b,c)\to(\,.\,)^{c+\mu-1}w_{i}(\,.\,;a,b,c+\mu)$.
These can all be obtained by rewriting fractional integral transformations for
${}_{2}F_{1}(a,b;c;\,.\,)$ of various types given in Section 4, namely types
$c+$; $a-$, $b-$, $c-$; $a-$; $a-$; $c-$; $a+$, $b+$, $c+$, respectively. Here
we list these formulas. Each formula is preceded by the formula number of the
formula in Section 4 to which it reduces:

$\displaystyle\eqref{10}\colon\
\int_{0<y/x<1}|y|^{c-1}w_{1}(y;a,b,c)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c)}{\Gamma(c+\mu)}|x|^{c+\mu-1}w_{1}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{10}\colon\
}{}\qquad\quad(x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,c>0,\;{\rm Re}\,\mu>0);$
(5.1) $\displaystyle\eqref{4}\colon\
\int_{-\infty}^{x}|y|^{c-1}w_{2}(y;a,b,c)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-c-\mu+1)}{\Gamma(a-c+1)}$
$\displaystyle\hphantom{\eqref{4}\colon\
}{}\qquad{}\times\frac{\Gamma(b-c-\mu+1)}{\Gamma(b-c+1)}\frac{\Gamma(2-c)}{\Gamma(2-c-\mu)}|x|^{c+\mu-1}w_{2}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{4}\colon\ }{}\qquad\quad(x\in(-\infty,1),\;{\rm
Re}\,(a-c+1),{\rm Re}\,(b-c+1)>{\rm Re}\,\mu>0);$ (5.2)
$\displaystyle\eqref{72}\colon\
\int_{y/x>1}|y|^{c-1}w_{3}(y;a,b,c)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-c-\mu+1)}{\Gamma(a-c+1)}|x|^{c+\mu-1}w_{3}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{72}\colon\
}{}\qquad\quad(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,(a-c+1)>{\rm
Re}\,\mu>0);$ (5.3) $\displaystyle\eqref{72}\colon\
\int_{y/x>1}|y|^{c-1}w_{4}(y;a,b,c)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(b-c-\mu+1)}{\Gamma(b-c+1)}|x|^{c+\mu-1}w_{4}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{72}\colon\
}{}\qquad\quad(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,(b-c+1)>{\rm
Re}\,\mu>0);$ (5.4) $\displaystyle\eqref{6}\colon\
\int_{x}^{\infty}y^{c-1}w_{5}(y;a,b,c)\frac{(y-x)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(b-c-\mu+1)}{\Gamma(b-c+1)}$
$\displaystyle\hphantom{\eqref{6}\colon\
}{}\qquad{}\times\frac{\Gamma(a-c-\mu+1)}{\Gamma(a-c+1)}\frac{\Gamma(a+b-c+1)}{\Gamma(a+b-c-\mu+1)}x^{c+\mu-1}w_{5}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{6}\colon\ }{}\qquad\quad(x\in(0,\infty),\;{\rm
Re}\,(b-c+1),{\rm Re}\,(a-c+1)>{\rm Re}\,(\mu)>0);$ (5.5)
$\displaystyle\eqref{7}\colon\
\int_{0<\frac{1-y}{1-x}<1}y^{c-1}w_{6}(y;a,b,c)\frac{|y-x|^{\mu-1}}{\Gamma(\mu)}\,dy$
$\displaystyle\hphantom{\eqref{7}\colon\
}{}\qquad=\frac{\Gamma(c-a-b+1)}{\Gamma(c-a-b+\mu+1)}x^{c+\mu-1}w_{6}(x;a,b,c+\mu)$
$\displaystyle\hphantom{\eqref{7}\colon\ }{}\qquad\quad\
(x\in(0,1)\cup(1,\infty),\;{\rm Re}\,(c-a-b+1)>0,\;{\rm Re}\,\mu>0).$ (5.6)

It is a straightforward exercise to list also the fractional integral
transformation formulas of the $w_{i}$ corresponding to the other seven types.
We do not list all these formulas here, but only give their essential
behaviour in the following table. Here an entry in $i$th row, $j$th column
(place $(i,j)$) tells us that the transformation formula for $w_{j}$ of type
given at $(i,1)$ can be reduced to the transformation formula for $w_{1}$ of
type given at $(i,j)$.

$w_{1}$ $w_{2}$ $w_{3}$ $w_{4}$ $w_{5}$ $w_{6}$ $c+$ $a-$, $b-$, $c-$ $a-$
$a-$ $c-$ $a+$, $b+$, $c+$ $a+$, $c+$ $a-$, $c-$ $a+$, $c+$ $a-$, $c-$ $a+$
$a+$ $a+$, $b+$, $c+$ $c-$ $a+$ $a+$ $a+$, $b+$, $c+$ $c-$ $a-$ $a-$ $a-$,
$b-$, $c-$ $c+$ $a-$, $c-$ $a+$, $c+$ $a+$ $a+$ $a+$, $b+$, $c+$ $c-$ $a+$,
$c+$ $a-$, $c-$ $a-$, $b-$, $c-$ $c+$ $a-$ $a-$ $a-$, $b-$, $c-$ $c+$ $a-$,
$c-$ $a+$, $c+$ $a-$, $c-$ $a+$, $c+$ $a-$ $a-$ $c-$ $a+$, $b+$, $c+$ $a+$
$a+$ $c+$ $a-$, $b-$, $c-$

The cases $c+$; $c-$; $a-$, $b-$, $c-$; $a+$, $b+$, $c+$ occur 5 times, the
cases $a+$, $c+$; $a-$, $c-$ 6 times, and the cases $a-$; $a+$ 8 times. That
these numbers are not all equal will be caused by the symmetry in $a$ and $b$
of the hypergeometric function.

Some more examples from the above table which we need later (because of
specialization to Euler type integral representations) are:

$\displaystyle\int_{0<\frac{y}{x}<1}|y|^{a-\mu-1}w_{2}(y;a,b,c)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(a-c-\mu+1)}{\Gamma(a-c+1)}|x|^{a-1}w_{2}(x;a-\mu,b,c)$
$\displaystyle\qquad(x\in(-\infty,0)\cup(0,1),\;{\rm Re}\,(a-c+1)>{\rm
Re}\,\mu>0),$ (5.7)
$\displaystyle\int_{\frac{y}{x}>1}w_{4}(y;a,b,c)\frac{|x-y|)^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(b-\mu)}{\Gamma(b)}w_{4}(x;a-\mu,b-\mu,c-\mu)$
$\displaystyle\qquad(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,b>{\rm
Re}\,\mu>0),$ (5.8)
$\displaystyle\int_{0<\frac{1-y}{1-x}<1}w_{6}(y;a,b,c)\frac{|x-y|^{\mu-1}}{\Gamma(\mu)}\,dy=\frac{\Gamma(c-a-b+1)}{\Gamma(c-a-b+\mu+1)}w_{6}(x;a-\mu,b-\mu,c-\mu)$
$\displaystyle\qquad(x\in(0,1)\cup(1,\infty),\;{\rm Re}\,(c-a-b+1)>0,\;{\rm
Re}\,\mu>0).$ (5.9)

##  6 Generalized Stieltjes transforms between solutions
of the hypergeometric differential equation

First we recall the proof of (3.22) as given by Karp & Sitnik [12, Lemma 2].
Expand for $|z|<1$ the left-hand side of (3.22) as

$\displaystyle\sum_{k=0}^{\infty}\frac{(a)_{k}}{k!}z^{k}\int_{0}^{1}t^{d+e-b-c-1}(1-t)^{b+k-1}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{d-c,e-c}{d+e-b-c};t\right)\,dt.$

By the limit case $x\uparrow 1$ of (1.5) this equals

$\displaystyle\sum_{k=0}^{\infty}\frac{(a)_{k}}{k!}z^{k}\,\frac{\Gamma(d+e-b-c)\Gamma(b+k)}{\Gamma(d+e-c+k)}\,\mbox{}_{2}F_{1}\\!\left(\genfrac{}{}{0.0pt}{}{d-c,e-c}{d+e-c+k};1\right).$

By the Gauss summation formula [1, Theorem 2.2.2] we get

$\displaystyle\sum_{k=0}^{\infty}\frac{(a)_{k}}{k!}z^{k}\frac{\Gamma(d+e-b-c)\Gamma(b+k)}{\Gamma(d+e-c+k)}\frac{\Gamma(c+k)\Gamma(d+e-c+k)}{\Gamma(d+k)\Gamma(e+k)},$

which is equal to the right-hand side of (3.22). All steps can be rigorously
justified because of the constraints in (3.22).

We now list the generalized Stieltjes transforms mapping a solution $w_{i}$ to
a solution $w_{j}$. They can all be obtained from the special case (3.23) of
(3.22) by change of parameters, change of integration variable, and
application of (2.2) and (2.3). In particular, (6.10) and (6.12) below are
simple rewritings of (3.23) by a change of parameters. Also, (6.1) below is a
rewriting of [8, 20.2(10)].

The formulas in the list are grouped in cases similar to the cases in Section
4. In all formulas the constraints are that $x$ is in an interval
$(-\infty,0)$, $(0,1)$ or $(1,\infty)$ which does not coincide with the
integration interval, and that the arguments of the three gamma functions in
the numerator on the right-hand side have positive real parts.

#### Case $\boldsymbol{c+}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{c-1}w_{1}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-c-\mu+1)\Gamma(b-c-\mu+1)\Gamma(c)}{\Gamma(a+b-c-\mu+1)\Gamma(1-\mu)}x^{c-1+\mu}w_{5}(x;a,b,c+\mu),$
(6.1) $\displaystyle\int_{1}^{\infty}y^{c-1}w_{6}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-c-\mu+1)\Gamma(b-c-\mu+1)\Gamma(c-a-b+1)}{\Gamma(2-c-\mu)\Gamma(1-\mu)}|x|^{c+\mu-1}w_{2}(x;a,b,c+\mu).$
(6.2)

#### Case $\boldsymbol{a+}$, $\boldsymbol{c+}$.

$\displaystyle\int_{0}^{1}y^{c-1}(1-y)^{b-c-\mu}w_{1}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(b-c-\mu+1)\Gamma(c)}{\Gamma(b-a-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c+\mu-1}|x-1|^{b-c}w_{4}(x;a+\mu,b,c+\mu),$
(6.3)
$\displaystyle\int_{1}^{\infty}y^{c-1}(y-1)^{b-c-\mu}w_{3}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(b-c-\mu+1)\Gamma(a-b+1)}{\Gamma(-c-\mu+2)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c+\mu-1}|x-1|^{b-c}w_{2}(x;a+\mu,b,c+\mu).$
(6.4)

#### Case $\boldsymbol{b+}$, $\boldsymbol{c+}$.

$\displaystyle\int_{0}^{1}y^{c-1}(1-y)^{a-c-\mu}w_{1}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-b-\mu+1)\Gamma(a-c-\mu+1)\Gamma(c)}{\Gamma(a-b-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c+\mu-1}|x-1|^{a-c}w_{3}(x;a,b+\mu,c+\mu),$
(6.5)
$\displaystyle\int_{1}^{\infty}y^{c-1}(y-1)^{a-c-\mu}w_{4}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma({-}b-\mu+1)\Gamma(a-c-\mu+1)\Gamma(b-a+1)}{\Gamma(-c-\mu+2)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c+\mu-1}|x-1|^{a-c}w_{2}(x;a,b+\mu,c+\mu).$
(6.6)

#### Case $\boldsymbol{a+}$, $\boldsymbol{b+}$, $\boldsymbol{c+}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{c-1}(1-y)^{a+b-c}w_{1}(x;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(-b-\mu+1)\Gamma(c)}{\Gamma(c-a-b-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times
x^{c+\mu-1}(1-x)^{a+b-c+\mu}w_{6}(x;a+\mu,b+\mu,c+\mu),$ (6.7)
$\displaystyle\int_{1}^{\infty}y^{c-1}(y-1)^{a+b-c}w_{5}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(-b-\mu+1)\Gamma(a+b-c+1)}{\Gamma(2-c-\mu)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c+\mu-1}(1-x)^{a+b-c+\mu}w_{2}(x;a+\mu,b+\mu,c+\mu).$
(6.8)

#### Case $\boldsymbol{a-}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{a-\mu-1}w_{4}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-\mu)\Gamma(a-c-\mu+1)\Gamma(b-a+1)}{\Gamma(a+b-c-\mu+1)\Gamma(1-\mu)}x^{a-1}w_{5}(x;a-\mu,b,c),$
(6.9) $\displaystyle\int_{0}^{1}y^{a-\mu-1}w_{6}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-c-\mu+1)\Gamma(a-\mu)\Gamma(c-a-b+1)}{\Gamma(a-b-\mu+1)\Gamma(1-\mu)}|x|^{a-1}w_{3}(x;a-\mu,b,c).$
(6.10)

#### Case $\boldsymbol{b-}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{b-\mu-1}w_{3}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(b-\mu)\Gamma(b-c-\mu+1)\Gamma(a-b+1)}{\Gamma(a+b-c-\mu+1)\Gamma(1-\mu)}x^{b-1}w_{5}(x;a,b-\mu,c),$
(6.11) $\displaystyle\int_{0}^{1}y^{b-\mu-1}w_{6}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(b-c-\mu+1)\Gamma(b-\mu)\Gamma(c-a-b+1)}{\Gamma(b-a-\mu+1)\Gamma(1-\mu)}|x|^{b-1}w_{4}(x;a,b-\mu,c).$
(6.12)

#### Case $\boldsymbol{a+}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{c-a-\mu-1}(1-y)^{a+b-c}w_{3}(y;a,b,c)(x-y)^{\mu-1}\,dy$
(6.13)
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(c-a-\mu)\Gamma(a-b+1)}{\Gamma(c-a-b-\mu+1)\Gamma(1-\mu)}x^{c-a-1}(1-x)^{a+b-c+\mu}w_{6}(x;a+\mu,b,c),$
$\displaystyle\int_{0}^{1}y^{c-a-\mu-1}(1-y)^{a+b-c}w_{5}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-a-\mu+1)\Gamma(c-a-\mu)\Gamma(a+b-c+1)}{\Gamma(b-a-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c-a-1}|1-x|^{a+b-c-\mu}w_{4}(x;a+\mu,b,c).$
(6.14)

#### Case $\boldsymbol{b+}$.

$\displaystyle\int_{-\infty}^{0}(-y)^{c-b-\mu-1}(1-y)^{a+b-c}w_{4}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-b-\mu+1)\Gamma(c-b-\mu)\Gamma(b-a+1)}{\Gamma(c-a-b-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times
x^{c-b-1}(1-x)^{a+b-c+\mu}w_{6}(x;a,b+\mu,c),$ (6.15)
$\displaystyle\int_{0}^{1}y^{c-b-\mu-1}(1-y)^{a+b-c}w_{5}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(-b-\mu+1)\Gamma(c-b-\mu)\Gamma(a+b-c+1)}{\Gamma(a-b-\mu+1)\Gamma(1-\mu)}$
$\displaystyle\qquad\quad{}\times|x|^{c-b-1}|1-x|^{a+b-c-\mu}w_{3}(x;a,b+\mu,c).$
(6.16)

#### Case $\boldsymbol{a-}$, $\boldsymbol{b-}$, $\boldsymbol{c-}$.

$\displaystyle\int_{-\infty}^{0}w_{2}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-\mu)\Gamma(b-\mu)\Gamma(2-c)}{\Gamma(a+b-c-\mu+1)\Gamma(1-\mu)}w_{5}(x;a-\mu,b-\mu,c-\mu),$
(6.17) $\displaystyle\int_{1}^{\infty}w_{6}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(a-\mu)\Gamma(b-\mu)\Gamma(c-a-b+1)}{\Gamma(c-\mu)\Gamma(1-\mu)}w_{1}(x;a-\mu,b-\mu,c-\mu).$
(6.18)

#### Case $\boldsymbol{a-}$, $\boldsymbol{c-}$.

$\displaystyle\int_{0}^{1}(1-y)^{a-\mu-1}w_{2}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-b-\mu)\Gamma(a-\mu)\Gamma(2-c)}{\Gamma(a-b-\mu+1)\Gamma(1-\mu)}|x-1|^{a-1}w_{3}(x;a-\mu,b,c-\mu),$
(6.19)
$\displaystyle\int_{1}^{\infty}(y-1)^{a-\mu-1}w_{4}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-b-\mu)\Gamma(a-\mu)\Gamma(b-a+1)}{\Gamma(c-\mu)\Gamma(1-\mu)}|1-x|^{a-1}w_{1}(x;a-\mu,b,c-\mu).$
(6.20)

#### Case $\boldsymbol{b-}$, $\boldsymbol{c-}$.

$\displaystyle\int_{0}^{1}(1-y)^{b-\mu-1}w_{2}(y;a,b,c)|x-y|^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-a-\mu)\Gamma(b-\mu)\Gamma(2-c)}{\Gamma(b-a-\mu+1)\Gamma(1-\mu)}|x-1|^{b-1}w_{4}(x;a,b-\mu,c-\mu),$
(6.21)
$\displaystyle\int_{1}^{\infty}(y-1)^{b-\mu-1}w_{3}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-a-\mu)\Gamma(b-\mu)\Gamma(a-b+1)}{\Gamma(c-\mu)\Gamma(1-\mu)}|1-x|^{b-1}w_{1}(x;a,b-\mu,c-\mu).$
(6.22)

#### Case $\boldsymbol{c-}$.

$\displaystyle\int_{-\infty}^{0}(1-y)^{a+b-c}w_{2}(y;a,b,c)(x-y)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-a-\mu)\Gamma(c-b-\mu)\Gamma(2-c)}{\Gamma(c-a-b-\mu+1)\Gamma(1-\mu)}|1-x|^{a+b-c+\mu}w_{6}(x;a,b,c-\mu),$
(6.23)
$\displaystyle\int_{1}^{\infty}(y-1)^{a+b-c}w_{5}(y;a,b,c)(y-x)^{\mu-1}\,dy$
$\displaystyle\qquad{}=\frac{\Gamma(c-a-\mu)\Gamma(c-b-\mu)\Gamma(a+b-c+1)}{\Gamma(c-\mu)\Gamma(1-\mu)}(1-x)^{a+b-c+\mu}w_{1}(x;a,b,c-\mu).$
(6.24)

We summarize the results of the above list in the following table. In the box
in row $w_{i}$ and column $w_{j}$ the type is given of the generalized
Stieltjes transform sending $w_{i}$ to $w_{j}$.

$w_{1}$ $w_{2}$ $w_{3}$ $w_{4}$ $w_{5}$ $w_{6}$ $w_{1}$ $b+$, $c+$ $a+$, $c+$
$c+$ $a+$, $b+$, $c+$ $w_{2}$ $a-$, $c-$ $b-$, $c-$ $a-$, $b-$, $c-$ $c-$
$w_{3}$ $b-$, $c-$ $a+$, $c+$ $b-$ $a+$ $w_{4}$ $a-$, $c-$ $b+$, $c+$ $a-$
$b+$ $w_{5}$ $c-$ $a+$, $b+$, $c+$ $b+$ $a+$ $w_{6}$ $a-$, $b-$, $c-$ $c+$
$a-$ $b-$

Due to the denominator factor $\Gamma(1-\mu)$ all generalized Stieltjes
transforms above become zero if $\mu$ is a positive integer satisfying the
constraints.

##  7 Euler type integral representations

Below we give the explicit Euler type integral representations which have form
(1.4) or (3.21).

#### Euler type integrals with integrand
$\boldsymbol{|x|^{1-c}|y|^{b-1}|1-y|^{-a}|x-y|^{c-b-1}}$.

$\displaystyle
w_{1}(x;a,b,c)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}|x|^{1-c}\int_{0<y/x<1}|y|^{b-1}(1-y)^{-a}|x-y|^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{1}(x;a,b,c)=}{}\ (x\in(-\infty,0)\cup(0,1),\;{\rm
Re}\,c>{\rm Re}\,b>0),$ (7.1) $\displaystyle
w_{2}(x;a,b,c)=\frac{\Gamma(2-c)}{\Gamma(a-c+1)\Gamma(1-a)}|x|^{1-c}\int_{1}^{\infty}y^{b-1}(y-1)^{-a}(y-x)^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{2}(x;a,b,c)=}{}\ (x\in(-\infty,0)\cup(0,1),\;{\rm
Re}\,(c-1)<{\rm Re}\,a<1),$ (7.2) $\displaystyle
w_{3}(x;a,b,c)=\frac{\Gamma(a-b+1)}{\Gamma(a-c+1)\Gamma(c-b)}|x|^{1-c}\int_{y/x>1}|y|^{b-1}|y-1|^{-a}|y-x|^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{3}(x;a,b,c)=}{}\
(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,(a-c+1)>0,\;{\rm Re}\,(c-b)>0),$
(7.3) $\displaystyle
w_{4}(x;a,b,c)=\frac{\Gamma(b-a+1)}{\Gamma(-a)\Gamma(b+1)}|x|^{1-c}\int_{0}^{1}y^{b-1}(1-y)^{-a}|x-y|^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{4}(x;a,b,c)=}{}\
(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,a<0,\;{\rm Re}\,b>-1),$ (7.4)
$\displaystyle
w_{5}(x;a,b,c)=\frac{\Gamma(a+b-c+1)}{\Gamma(a-c+1)\Gamma(b)}x^{1-c}\int_{-\infty}^{0}(-y)^{b-1}(1-y)^{-a}(x-y)^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{5}(x;a,b,c)=}{}\ (x\in(0,\infty),\;{\rm
Re}\,b>0,\;{\rm Re}\,(a-c+1)>0),$ (7.5) $\displaystyle
w_{6}(x;a,b,c)=\frac{\Gamma(c-a-b+1)}{\Gamma(1-a)\Gamma(c-b)}x^{1-c}\int_{0<\frac{1-y}{1-x}<1}y^{b-1}(1-y)^{-a}|x-y|^{c-b-1}\,dy$
$\displaystyle\hphantom{w_{6}(x;a,b,c)=}{}\ (x\in(0,1)\cup(1,\infty),\;{\rm
Re}\,(c-b)>0,\;{\rm Re}\,a<1).$ (7.6)

#### Euler type integral transformations with integrand
$\boldsymbol{|y|^{a-c}|1-y|^{c-b-1}|x-y|^{-a}}$.

$\displaystyle
w_{1}(x;a,b,c)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{1}^{\infty}y^{a-c}(y-1)^{c-b-1}(y-x)^{-a}\,dy$
$\displaystyle\hphantom{w_{1}(x;a,b,c)=}{}\ (x<1,\;{\rm Re}\,c>{\rm
Re}\,b>0),$ (7.7) $\displaystyle
w_{2}(x;a,b,c)=\frac{\Gamma(2-c)}{\Gamma(a-c+1)\Gamma(1-a)}\int_{0<y/x<1}|y|^{a-c}(1-y)^{c-b-1}|x-y|^{-a}\,dy$
$\displaystyle\hphantom{w_{2}(x;a,b,c)=}{}\ (x\in(-\infty,0)\cup(0,1),\;2>{\rm
Re}\,(a+1)>{\rm Re}\,c),$ (7.8) $\displaystyle
w_{3}(x;a,b,c)=\frac{\Gamma(a-b+1)}{\Gamma(a-c+1)\Gamma(c-b)}\int_{0}^{1}y^{a-c}(1-y)^{c-b-1}|x-y|^{-a}\,dy$
$\displaystyle\hphantom{w_{3}(x;a,b,c)=}{}\
(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,(a+1)>{\rm Re}\,c>{\rm Re}\,b),$
(7.9) $\displaystyle
w_{4}(x;a,b,c)=\frac{\Gamma(b-a+1)}{\Gamma(-a)\Gamma(b+1)}\int_{y/x>1}|y|^{a-c}|y-1|^{c-b-1}|y-x|^{-a}\,dy$
$\displaystyle\hphantom{w_{4}(x;a,b,c)=}{}\
(x\in(-\infty,0)\cup(1,\infty),\;{\rm Re}\,a<1,\;{\rm Re}\,b>0),$ (7.10)
$\displaystyle
w_{5}(x;a,b,c)=\frac{\Gamma(a+b-c+1)}{\Gamma(a-c+1)\Gamma(b)}\int_{-\infty}^{0}(-y)^{a-c}(1-y)^{c-b-1}(x-y)^{-a}\,dy$
$\displaystyle\hphantom{w_{5}(x;a,b,c)=}{}\ (x\in(0,\infty),\;{\rm
Re}\,b>0,\;{\rm Re}\,(a+1)>{\rm Re}\,c),$ (7.11) $\displaystyle
w_{6}(x;a,b,c)=\frac{\Gamma(c-a-b+1)}{\Gamma(1-a)\Gamma(c-b)}\int_{0<\frac{1-y}{1-x}<1}|y|^{a-c}|1-y|^{c-b-1}|x-y|^{-a}\,dy$
$\displaystyle\hphantom{w_{6}(x;a,b,c)=}{}\ (x\in(0,1)\cup(1,\infty),\;{\rm
Re}\,(c-b)>0,\;{\rm Re}\,a<1).$ (7.12)

The integrals in (7.1), (7.3), (7.6) (for $w_{1}$, $w_{3}$, $w_{6}$) and in
(7.8), (7.10), (7.12) (for $w_{2}$, $w_{4}$, $w_{6}$) are of fractional
integral type. Formulas (7.3), (7.8), (7.10) and (7.12) can be reduced to
(1.1) by a change of integration variable, while (7.1) is a rewritten version
of (1.1). Each of the six formulas (7.1)–(7.6) is paired with one of the six
formulas (7.7)–(7.12) by a transformation of integration variable which
involves $x$. The pairing is as follows:

$\displaystyle\eqref{23}\ \leftrightarrow\ \eqref{30},\qquad$
$\displaystyle\eqref{24}\ \leftrightarrow\ \eqref{31},\qquad$
$\displaystyle\eqref{25}\ \leftrightarrow\ \eqref{22},$
$\displaystyle\eqref{26}\ \leftrightarrow\ \eqref{29},\qquad$
$\displaystyle\eqref{27}\ \leftrightarrow\ \eqref{32},\qquad$
$\displaystyle\eqref{28}\ \leftrightarrow\ \eqref{18}.$

In 1874 Letnikov [14, (11), (13), (16), (17), (21), (22)] (see also Sostak
[23, Section 4]) already gave the Euler type integral representations (7.8),
(7.12), (7.10), (7.1), (7.6), (7.3), respectively, as solutions of the
hypergeometric differential equation [14, p. 115, (A)]. In order to arrive at
these results he essentially considered the fractional integral operator as a
transmutation operator with respect to the hypergeometric differential
operator.

int. rep. transform from to case (7.1) (5.1) $w_{1}$ $w_{1}$ $c+$ (7.2) (6.4)
$w_{3}$ $w_{2}$ $a+$, $c+$ (7.3) (5.3) $w_{3}$ $w_{3}$ $c+$ (7.4) (6.3)
$w_{1}$ $w_{4}$ $a+$, $c+$ (7.5) (6.1) $w_{1}$ $w_{5}$ $c+$ (7.6) (5.6)
$w_{6}$ $w_{6}$ $c+$ (7.7) (6.20) $w_{4}$ $w_{1}$ $a-$, $c-$ (7.8) (5.7)
$w_{2}$ $w_{2}$ $a-$ (7.9) (6.10) $w_{6}$ $w_{3}$ $a-$ (7.10) (5.8) $w_{4}$
$w_{4}$ $a-$, $b-$, $c-$ (7.11) (6.9) $w_{4}$ $w_{5}$ $a-$ (7.12) (5.9)
$w_{6}$ $w_{6}$ $a-$, $b-$, $c-$

The above table gives for each Euler type integral representation in the first
column the transformation formula in the second column from which it can be
obtained by specialization of parameters ($b=c$ in the first six rows and
$a=-1$ in the last six rows). The transformation formula sends $w_{i}$ in the
third column to $w_{j}$ in the fourth column. Its case is given in the fifth
column. Formula (7.5) is essentially the same as [8, 14.4(9)].

##  8 Generalized Stieltjes transform
and fractional integral transform combined

As observed in [8, p. 213], generalized Stieltjes transforms of different
order are connected with each other by fractional integration. Formula (8.1)
below was essentially given there, and later given with proof in [10, Theorem
9]. Some related identities can also be proved:

######  Proposition 8.1.

Let ${\rm Re}\,(1-\nu)>{\rm Re}\,\mu>0$. Assume that $f\in L_{\rm
loc}^{1}((m,M))$ and that the integrals on the right-hand side of the four
identities below converge absolutely. Then

$\displaystyle\int_{-\infty}^{x}\left(\int_{m}^{M}f(z)\frac{\Gamma(1-\nu)}{(z-y)^{1-\nu}}\,dz\right)\frac{(x-y)^{\mu-1}}{\Gamma(\mu)}\,dy=\int_{m}^{M}f(z)\frac{\Gamma(1-\mu-\nu)}{(z-x)^{1-\mu-\nu}}\,dz\quad(x<m),\\!\\!\\!$
(8.1)
$\displaystyle\int_{x}^{\infty}\left(\int_{m}^{M}f(z)\frac{\Gamma(1-\nu)}{(y-z)^{1-\nu}}\,dz\right)\frac{(y-x)^{\mu-1}}{\Gamma(\mu)}\,dy=\int_{m}^{M}f(z)\frac{\Gamma(1-\mu-\nu)}{(x-z)^{1-\mu-\nu}}\,dz\quad(M<x),\\!\\!\\!$
(8.2)
$\displaystyle\int_{m}^{\infty}\left(\int_{m}^{y}f(z)\frac{(y-z)^{\mu-1}}{\Gamma(\mu)}\,dz\right)\frac{\Gamma(1-\nu)}{(x-y)^{1-\nu}}\,dy=\int_{m}^{\infty}f(z)\frac{\Gamma(1-\mu-\nu)}{(z-x)^{1-\mu-\nu}}\,dz\quad(x<m),$
(8.3)
$\displaystyle\int_{-\infty}^{M}\left(\int_{y}^{M}f(z)\frac{(z-y)^{\mu-1}}{\Gamma(\mu)}\,dz\right)\frac{\Gamma(1-\nu)}{(y-x)^{1-\nu}}\,dy=\int_{-\infty}^{M}f(z)\frac{\Gamma(1-\mu-\nu)}{(x-z)^{1-\mu-\nu}}\,dz\quad(x>M).\\!\\!\\!\\!$
(8.4)

The proofs are immediate, by the Fubini theorem and by a version [18,
(5.12.3)] of the beta integral. Furthermore, (8.2) is an immediate consequence
of (8.1), and similarly (8.4) of (8.3).

Examples of formulas (8.1)–(8.4) can be found by combining suitable fractional
integral formulas in Section 5 with suitable generalized Stieltjes transform
formulas in Section 6. For instance:

  * •

In (8.1) let $m=1$, $M=\infty$, $f(z)=z^{c-1}w_{6}(z;a,b,c)$ and use (5.2) and
(6.2).

  * •

In (8.2) let $m=-\infty$, $M=0$, $f(z)=(-z)^{c-1}w_{1}(z;a,b,c)$ and use (5.5)
and (6.1).

  * •

In (8.3) let $m=1$, $f(z)=z^{c-1}w_{6}(z;a,b,c)$ and use (5.6) and (6.2).

  * •

In (8.4) let $M=0$, $f(z)=(-z)^{c-1}w_{1}(z;a,b,c)$ and use (5.1) and (6.1).

### Acknowledgements

I am very grateful to Sergei Sitnik for his comments, in particular about
Letnikov’s paper [14] from 1874. Thanks also to Dmitry Karp for helpful
comments. Furthermore, the paper took profit from comments and lists of typos
in referees’ reports.

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##  Appendix A Shear and Redshift Bias Parameters used Prior to Unblinding the
Analysis

A final technical improvement to the priors on $m$ and $\delta\bar{z}$ was
completed in parallel with the unblinding of the DES Y3 3x2pt cosmology
analysis. These updates were not expected to impact the analysis in a
significant way. The original chains at unblinding used a different (but very
similar) set of $m$ and $\delta\bar{z}$ priors. Gaussian priors on the per-bin
$m$ were marginalized over with mean, $\mu$ and width, $\sigma$ for the four
photo-$z$ bins given by
$(\mu,\sigma)=\left[(-0.01044,0.00576),(-0.01579,0.00414),(-0.02489,0.00532),(-0.03802,0.00801)\right]$.
The mean of the $n_{\gamma}(z)$ samples generated using our fiducial
correction was used, with a Gaussian $\delta\bar{z}$ prior estimated from
their statistical scatter only, with width
$\left[0.018,0.013,0.006,0.013\right]$.

DES Collaboration et al. (2020) describe the unblinding criteria and procedure
in detail, and conclude that updating the priors had negligible impact on the
final cosmological constraints, and would not have affected the unblinding
criteria (and therefore the decision to unblind).

##  Appendix B Simulation Validation Tests

One of the main concerns when working with the complex simulation suite
presented in this work is that the simulations themselves could contain bugs
which would cause non-zero multiplicative or additive biases. To guard against
this possibility, we tested the performance of our pipelines and the
simulations in configurations where previous studies have demonstrated
sub-$0.1\%$ performance.

In these simulations, all objects have exponential light profiles. We fix the
objects to have the same magnitude in each band, but different simulations
vary the magnitude overall. The galaxies have a half-light radius that is
either fixed to 0.5$\arcsec$ or allowed to vary according to the input
catalogs for the more realistic simulations, although in the latter case, we
limit the half-light radius to at most 0.8$\arcsec$ in order to prevent
adjacent objects from overlapping. We test both a Gaussian PSF with a FWHM of
0.9$\arcsec$ and the smoothed Piff PSF models. We also vary whether or not we
apply the bad pixel masks from the data to the simulations (and then correct
for them). See Sevilla-Noarbe et al. (2020) and Gatti & Sheldon et al.,
(2020b) for more details on the bad pixel masks and how our measurement codes
correct for them. Further, we vary whether or not we perform pixel-level
outlier rejection on the individual postage stamps for each object. Finally,
we also vary a detail of how Metacalibration handles the computation of the
PSF it uses in its reconvolution step. These variants are denoted internally
as symmetrize or gauss-fit. The simulation configurations we tested are listed
in Table 6.

The measurements for $m$ and $c$ from these simulations are in the last two
columns of Table 6. We find that in the simplest case (configuration 1), we
recover the applied shear to better than a part in a thousand with no evidence
of any additive bias. The same conclusion holds for configurations 2, 3, and
4a which vary the galaxy sizes, the PSF model, and the Metacalibration
reconvolution PSF treatment. Comparing configurations 4a and 4, we find that
galaxy size variation coupled with the symmetrize Metacalibration PSF
treatment introduces a small additive error in $g2$, but one that is too small
to impact cosmology. Configuration 5, which introduces pixel masking effects
relative to configuration 4, demonstrates that our masking corrections cause a
small $\sim-0.2\%$ multiplicative bias. This effect is well below our
requirements and is subdominant relative to detection and/or blending, as
described above. Configurations 6, 7, 8, and 9 use the pixel-level outlier
rejection and employ a series of fainter objects from magnitude 17 to
magnitude 20.5. We find that extremely bright objects exhibit some non-trivial
multiplicative bias in this case. This effect was traced back to Poisson noise
from the object’s flux falsely triggering the pixel-level outlier rejection
(which only considered the sky noise in its threshold). In the data, these
bright objects are extremely rare. Further, this effect is correctly modeled
in our simulations for any objects that do eventually pass the cuts we employ
on the shear catalogs.

config. # | gal. mag. | gal. size | PSF model | masking effects | outlier rej. | mcal PSF | m ($\times 10^{-3}$) | c ($\times 10^{-5}$)
---|---|---|---|---|---|---|---|---
1 | 17 | 0.5$\arcsec$ | Gaussian | no | no | gauss-fit | g1: $0.225\pm 0.005$ | $0.1\pm 0.1$
| g2: $0.440\pm 0.004$ | $-0.1\pm 0.1$
2 | 17 | variable | Gaussian | no | no | gauss-fit | g1: $0.314\pm 0.007$ | $-0.2\pm 0.2$
| g2: $0.415\pm 0.007$ | $-0.1\pm 0.02$
3 | 17 | variable | Gaussian | no | no | symmetrize | g1: $0.381\pm 0.009$ | $0.1\pm 0.2$
| g2: $0.383\pm 0.009$ | $-0.4\pm 0.5$
4a | 17 | variable | Piff | no | no | gauss-fit | g1: $0.219\pm 0.025$ | $-0.1\pm 0.3$
| g2: $0.294\pm 0.127$ | $0.1\pm 0.3$
4 | 17 | variable | Piff | no | no | symmetrize | g1: $0.405\pm 0.075$ | $0.2\pm 0.3$
| g2: $0.414\pm 0.014$ | $1.4\pm 0.4$
5 | 17 | variable | Piff | yes | no | symmetrize | g1: $-1.95\pm 0.30$ | $0.6\pm 0.4$
| g2: $-1.94\pm 0.29$ | $6.8\pm 1.0$
6 | 18 | variable | Piff | no | yes | symmetrize | g1: $8.56\pm 0.16$ | $0.0\pm 0.9$
| g2: $8.70\pm 0.13$ | $-9.5\pm 1.0$
7 | 18.75 | variable | Piff | no | yes | symmetrize | g1: $3.79\pm 0.33$ | $-2.0\pm 2.3$
| g2: $3.78\pm 0.13$ | $-2.3\pm 1.9$
8 | 19.5 | variable | Piff | no | yes | symmetrize | g1: $2.44\pm 0.16$ | $-2.8\pm 2.4$
| g2: $2.50\pm 0.11$ | $-1.0\pm 2.6$
9 | 20.5 | variable | Piff | no | yes | symmetrize | g1: $0.89\pm 0.29$ | $-5.9\pm 6.1$
| g2: $0.56\pm 0.26$ | $0.1\pm 6.6$
Table 6:  Validation simulation configurations and results. All simulations
place the galaxies on a grid and use “true detection” where the shear
measurement code is fed the true position of the galaxy in the image. We then
run variations with and without bad pixel masking and corrections ("masking
effects"), pixel-level outlier rejection ("outlier rej."), the galaxy and PSF
models ("gal. mag", "gal. size", and "PSF model"), and the Metacalibration
treatment of the PSF model ("mcal PSF"). The last two columns list the
multiplicative and additive biases we find in these simulation setups. See the
text in Appendix B for details.

##  Appendix C Piff PSF Model Smoothing for Simulations

The simulations in this work use the observed Piff PSF models. We found a bug
in the process of doing the DES Y3 analysis, detailed in Jarvis et al. (2020,
§3.2), which causes some images of the Piff PSF model to have what is
effectively extra background noise. In tests of the simulations, we found that
when using the model images as the true PSF for the simulation, sometimes this
extra noise caused shear recovery biases with Metacalibration even in cases
where that method is known to work at extremely high precision. In order to
avoid any potential problems in the simulations presented here, we applied the
following smoothing procedure to the Piff PSF models. For each single-epoch
image, we fit the rendered image of the Piff model with an ngmix turb model
(which is an approximation to a Kolmogorov-like PSF for a long exposure taken
through atmospheric turbulence). This fit is done over a grid of locations on
the CCD and the parameters of it are interpolated. We then use these
interpolated fit parameters to draw the image of the PSF model at any location
on the CCD. We include a small correction for the size of pixel in the fit
parameters in order to more closely match the output PSF sizes when the PSF is
drawn.

Figure 13 shows the differences in the shapes and FWHM for the raw Piff models
and the rendered smooth models. To make these histograms, we selected $10^{4}$
CCDs at random, fit our smoothed, interpolated models, and then drew images of
both the smooth model and the interpolated one at 10 random points per CCD. We
find an extremely small negative bias in the FWHM, but otherwise the models
provide a good approximation to the true PSF size and shape variations.

Figure 13: Performance of the smoothed Piff PSF models. Each panel from left
to right shows a histogram of the differences in the shape ($g_{1}$ or
$g_{2}$) or FWHM of the smoothed PSF model versus the underlying Piff PSF
model.

##  Appendix D The Effect of Smooth Response Weighting on Redshift
Distributions

As previously noted in Section 2.3, the redshift distributions are weighted by
the METACALIBRATION response. It was found in general that the noise in
response measurement could result in unrealistic redshift distributions, where
negatively measured responses could lead to an unphysical negative $N(z)$ at
several points. Noise in response measurement then, tracks directly into N(z)
scatter.

The DES Year 3 analysis resolves this problem by smoothing the same two sets
of weights (shear and response) over a log grid in signal to noise mcal_s2n_r
and the size ratio with respect to the point spread function,
$T_{\mathrm{mcal}}/T_{\mathrm{psf}}$. One such grid, generated from the mesh
average in the data is plotted in Fig. 14. For each detection in the image
simulations, we assign it to a cell on this grid of SNR and size ratio, and
take the mean response of all detections in that cell. This mean response (or
_smooth response_) is then used when weighting the redshift distribution.

For this weighting scheme to be valid, we tested the shift in mean redshift
for each tomographic bin in the image simulations from the reweighting, as
this data set has a very well understood input redshift distribution. The
shift applied as a result of smoothing in each bin was on average 0.0025 in
mean redshift, with the first bin being the most affected with a shift of
0.004, and the third being the least affected at 0.0002. Response weighting as
a whole shifts the mean redshift in each bin by an order of magnitude larger,
so the majority of the effect is preserved. In conclusion, given the reduction
in $N(z)$ scatter achieved with a smooth response weight and the small
amplitude of bias ($\sim 10^{-3}$), this approximation can be made when
evaluating the Year 3 data.

Figure 14: The smooth response grid generated from the DES Y3 catalog shows
correlation between mean response and size ratio, and mean response and signal
to noise. This 20 x 20 log-spaced grid supplied the smoothed weighting scheme
evaluated in App. D.

##  Appendix E Affiliations

1 Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, UK
2 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
3 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
4 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford,
CA 94305, USA
5 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
6 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
7 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA
8 Department of Physics, Duke University Durham, NC 27708, USA
9 Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA
10 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
11 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA
12 Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK
13 Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
14 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,
85748 Garching, Germany
15 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany
16 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
W. Green Street, Urbana, IL 61801, USA
17 National Center for Supercomputing Applications, 1205 West Clark St.,
Urbana, IL 61801, USA
18 Department of Physics, University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford OX1 3RH, UK
19 Département de Physique Théorique and Center for Astroparticle Physics,
Université de Genève, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland
20 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
21 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
22 Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
23 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
24 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
25 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093
Zurich, Switzerland
26 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
27 Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile
28 Departamento de Física Matemática, Instituto de Física, Universidade de São
Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
29 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
30 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
31 Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France
32 Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
33 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
34 Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain
35 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
36 Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
37 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
38 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
39 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
40 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
41 Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679
Munich, Germany
42 Department of Astronomy/Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA
43 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
44 Kavli Institute for Cosmological Physics, University of Chicago, Chicago,
IL 60637, USA
45 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049
Madrid, Spain
46 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
47 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
48 Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
49 School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
50 Department of Physics, The Ohio State University, Columbus, OH 43210, USA
51 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
52 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, and Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA
53 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA
54 Radcliffe Institute for Advanced Study, Harvard University, Cambridge, MA
02138
55 Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
56 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
57 Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390
58 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
59 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831
60 Department of Physics and Astronomy, Pevensey Building, University of
Sussex, Brighton, BN1 9QH, UK