- Open Access
Remark on certain transformations for multiple hypergeometric functions
© Gaboury and Tremblay; licensee Springer. 2014
- Received: 14 January 2014
- Accepted: 10 April 2014
- Published: 6 May 2014
In this paper, we provide many new general transformations for multiple hypergeometric functions. These transformations can be viewed as generalizations of some of those obtained recently by Wei et al. (Adv. Differ. Equ. 2013:360, 2013). We obtain these transformations by using the fractional calculus method which is a more general method than the beta integral method.
MSC:26A33, 33C20, 33C05.
- fractional derivatives
- Appell functions
- Srivastava function
- beta integral
- multiple hypergeometric series
For the numerous conditions of convergence for this function, the reader is referred to .
and abbreviates the array of A parameters with similar interpretations for , , , and so on.
The so-called beta integral method consists essentially of integral from 0 to 1 expressions which contain terms in the form to obtain new transformations formulas.
The aim of this paper is to present many new general transformations for multiple hypergeometric functions. These transformations can be viewed as generalizations of some of those obtained recently by Wei et al. . All these transformations are obtained by using a fractional calculus operator based on the Pochhammer contour integral. In Section 2, we give the representation of the fractional derivatives based on the Pochhammer contour of integration. Section 3 is devoted to the fractional calculus operator introduced by Tremblay . Finally, in Section 4, we present the several transformations involving multi-variable hypergeometric functions.
2 Pochhammer contour integral representation for fractional derivative and a new generalized Leibniz rule
This allows one to modify the restriction to . Another used representation for the fractional derivative is the one based on the Cauchy integral formula widely used by Osler [19–22]. These two representations have been used in many interesting research papers. It appears that the less restrictive representation of fractional derivative according to parameters is the Pochhammer contour definition introduced in [15, 23] (see also [24–28]).
Remark 2.2 In Definition 2.1, the function must be analytic at . However, it is interesting to note here that we could also allow to have an essential singularity at , and Equation (2.3) would still be valid.
Remark 2.3 The Pochhammer contour never crosses the singularities at and in (2.3), then we know that the integral is analytic for all p and for all α and for z in . Indeed, the only possible singularities of are , and which can directly be identified from the coefficient of the integral (2.3). However, integrating by parts N times the integral in (2.3) by two different ways, we can show that , and are removable singularities (see ).
but adopting the Pochhammer-based representation for the fractional derivative this last restriction becomes p not a negative integer.
- (4)Elementary cases(3.6)(3.7)
- (5)Useful cases(3.8)(3.9)(3.10)
It is worthy to mention that operator has a lot more interesting properties and applications. Tremblay introduced this operator in order to deal with special functions more efficiently and to facilitate the obtention of new relations such as hypergeometric transformations.
This relation shows, in fact, that the so-called beta integral method consists in a fractional derivative evaluated at the point .
In this section, we apply the fractional calculus operator to certain transformations involving multi-variable hypergeometric functions in order to obtain new transformations more general than those obtained by means of the beta integral method. Many special cases are also computed.
This completes the proof. □
Let us give a special case of Theorem 4.1 in which we recover a result given recently by Wei et al. [, Theorem 1].
gives the result. □
Rewriting (4.11) into the form of (1.2) leads to the desired result. □
Proof Putting in Theorem 4.3, using the Gauss summation formula (4.7) and making elementary simplifications yields the result. □
the result follows easily after simple calculations. □
and applying successively the operator and the operator on both sides of (4.17) gives the result. □
Setting in Theorem 4.6 and using twice the Gauss summation formula (4.7) leads to a result given by Wei et al. [, p.5], that is,
Applying the operator on both sides of (4.20) in a similar way as in the proofs of the previous theorems gives the result. □
If we set in Theorem 4.8, we obtain the following corollary which has been given by Wei et al. [, p.8].
Summing the hypergeometric function in the left member of (4.22) with the help of the Gauss summation formula (4.7) gives the result. □
Note that this result has been given recently by Wei et al. [, p.8].
Let us complete this paper by giving one last transformation.
if we apply the operator on both sides (4.24), the result follows easily after simple calculations. □
Using the Gauss summation theorem (4.7), the result follows easily. □
The previous corollary has been given by Wei et al. [, p.11].
It is important to mention here that the fractional calculus operator used in this paper can provide many very general transformation formulas involving hypergeometric functions of several variables. Tremblay  obtained many new transformation formulas with the help of this fractional calculus operator. A paper dealing with these new relations is in preparation.
The authors wish to thank the referees for valuable suggestions and comments.
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