Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators

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.


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 [6, 2.8(20)-(27)] and [12, ( §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 and that then essentially the same proof should also yield that is a solution of the hypergeometric differential equation if m, M and x are as follows: This means that m and M in (1.3) are two consecutive points of singularity of the integrand.Indeed, we will see that in all listed cases the right-hand side of (1.3) equals a constant multiple of one of the six explicit solutions w j [12, §15.10(ii)] of the hypergeometric differential equation.Some of these Euler type integral representations 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]:It will turn out that (1.4), 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 (1.2), acts as a so-called transmutation operator.For instance, in connection with (1.4) we have the transmutation formula 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 [10], namely as an operator A, often an integral operator, intertwining between two differential operators L 1 and L 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.
The cases of (1.3) where m and M are not equal to x are variants of the generalized Stieltjes transform, introduced by Widder [16, Section 8].We consider this as a transform sending f to g of the form where (m, M ) is (−∞, 0) or (0, 1) or (1, ∞), where x is on R outside the integration interval, and where the function y → |y| µ−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 [7].There [7, 14.4(9)] is essentially the case (m, M ) = (−∞, 0) of (1.3), while [7, 20.2(10)] is a generalized Stieltjes transform sending a 2 F 1 to a 2 F 1 .A formula by Karp & Sitnik [8,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 µ = 0 of (1.6) is essentially the classical Stieltjes transform.It will not change the parameters of the hypergeometric solutions.A well-known example of this case is [15, (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 [14], [9].They refer to an earlier proof in this spirit by Rainville [13,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 (1.4).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.

Preliminaries about hypergeometric functions
The Gauss hypergeometric function [6, Chapter 2], [1, Chapter 2], [12,Chapter 15] is defined by its power series ) 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 ).An elementary special case of the hypergeometric function is In the case of generic parameters there are essentially six different explicit solutions of the Gauss differential equation L a,b,c f = 0 [6, Section 2.9], [12, §15.10(ii)].These are one-valued analytic functions on the complex plane with a suitable real interval as cut: ) ) ) ) (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 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. ) (2.16) 3 The main ideas illustrated in a special case

Transmutation property of a differentiation operator
In [6, 2.8(20)-( 27)] or [12, ( §15.5)] there is a list of eight parameter changing n-th derivative formulas for Gauss hypergeometric functions.One of these is [12, (15.5.4)]: 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 where 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: 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, . . ., 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), This is equivalent to the prototypical differentiation formula for the hypergeometric function [4, (15.5.4)]: Corresponding to each of the eight n-th derivative formulas [12, ((15.5.2)-(15.5.9))] one can write down a transmutation formulas 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.
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)]): This suggests identities involving ).These can indeed be given, but sometimes one needs another branch cut for the power factor than chosen in (2.5)-(2.10).

Transmutation property of a fractional integral operator
Both Riemann-Liouville and Weyl fractional integrals will occur in our formulas.See for instance [7, pp. 181-183] or [11].
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 Re c > n.We obtain Note that there are no other terms because, by our assumption, y c−m vanishes at y = 0 for m = 1, . . ., n. Formula (3.10) has a fractional extension for n complex with Re c > Re n > 0. In rewritten form this is Bateman's fractional integral formula (1.4), which is often written in the form (3.12) We also need (straightforward by integration by parts): Lemma 3.2.We have: If m = 0 or M = 0 respectively m = 1 or M = 1 then the side conditions for x = 0 or x = 1 can be relaxed to f (0)g(0) = 0 respectively f (1)g(1) = 0.If m = −∞ or M = ∞ then the side conditions at ±∞ should be replaced by xf (x)g(x) → 0, In particular, L 1−a,1−b,2−c is the formal adjoint of L a,b,c .
Proof of (1.4) by (3.12).First assume 0 Here we applied (3.12) and Lemma 3.2.Note that the conditions for interchanging L a,b,c+µ;x • x 1−c−µ with x 0 .dy and for applying Lemma 3.2 are satisfied.Because this is analytic in x at x = 0, and for x = 0 it takes the value Thus, by the characterization of the hypergeometric function as solution of the hypergeometric differential equation, we have proved (3.12) for Re µ > 2, 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 (1.5).
Formally, the proof of (1.5) can be extended to showing that Here m or M may be equal to x, but not necessarily, and y and x − y should not change sign for y ∈ [m, M ).This becomes rigorous if f , c and µ are such that the second and fourth 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.
In this paper many explicit cases of (3.15) will be given, where f = w i for some i and g = const.w j for some j, with j = i if m or M equals x.As an example it can be derived that This is the fractional generalization of the iteration of (3.6).It can be equivalently written as This is the fractional generalization of the iteration of (3.7).A different proof will be given in th next section.
Another explicit case of (3.15) which we will meet is 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.

Euler type integral representations
When we replace c, µ by b, c − b in (1.4) 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 The right-hand side of (1.1) is annihilated by L a,b,c;x .This is a consequence of the transmutation property (1.5), by which L a,b,c;x acting on the right-hand side of (1.1) is equal to const.The more general transmutation property (3.14), considered with c, µ replaced by b, c − b, suggests that each w j has an Euler type integral representation (1.3).This is indeed the case, as we already briefly indicated after (1.3).
We can write (3.19) also equivalently as a generalized Stieltjes transform 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.

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 [16, Section 8] the generalized Stieltjes transform sends a suitable measure α or function φ (with dα(t) = φ(t) dt) to a function f analytic on C\(−∞, 0]: The special case ρ = 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 (1.6).
Transforms of generalized Stieltjes type have transmutation properties.For instance, from (3.14) we have, associated with (3.18), the intertwining property Noteworthy is the case µ = 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 [8, Lemma 2] proved the following formula: 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.If moreover d = a it sends a 2 F 1 to a 2 F 1 :

The eight fractional integral transformations of the Gauss hypergeometric function
Each of the eight n-th derivative formulas in [12, ( §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:

Transmutation formulas
We gave already (3.12),which gives rise to the transmutation formula (1.5) and suggests transmutation formulas (3.14), and by which a proof of (4.1) can be given.Now we list formulas similar to (3.12) for all cases.They can be obtained by straightforward computation, possibly using computer algebra.
c+ : a+, c+ a+, b+, c+ : a− : a+ : a−, c− : 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) → ( . ) c+µ−1 w i ( .; a, b, c + µ).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.
(4.1) : x c+µ−1 w 5 (x; a, b, c + µ) 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 ith row, jth 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).Some more examples from the above table which we need later (because of specialization to Euler type integral representations) are: 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 [7, 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 (−∞, 0), (0, 1) or (1, ∞) 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.
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 .(5.9) w 6 w 6 a−, b−, c− Formula (7.5) is essentially the same as [7, 14.4(9)].

Generalized Stieltjes transform and fractional integral transform combined
As observed in [7, 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 some related identities can also be proved: The proofs are immediate, by the Fubini theorem and by a version [12, (5.12.3)] of the beta integral.Furthermore, (8.2) is an immediate consequence of (8.1), and similarly (8.4) of (8.3).

x 1−c x 0 y
b−1 L a,b,b;y (1 − y) −a (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 L a,b,b;y = y d dy + b • (1 − y) d dy − a .
(3.20)It will turn out that, more generally and analogous to (1.3),g(x) = 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 (3.20).
C\(−∞, 1], Re b, Re c, 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).
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.