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Remark on certain transformations for multiple hypergeometric functions
Advances in Difference Equations volume 2014, Article number: 126 (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.
The largely investigated generalized hypergeometric function with p numerator parameters such that () and q denominator parameters such that (; ) is defined by (see, for example [, Chapter 4]; see also [, pp.71-72])
and denotes the Pochhammer symbol defined, in terms of the Gamma function, by
Multi-variable hypergeometric functions and their reduction formulas have also been largely investigated (for example, see ). Let us recall the general definition of the double hypergeometric function given by Srivastava and Panda [, p.423, Eq. (26)]. Let denotes the sequence of parameters , and let nonnegative integers define the Pochhammer symbol . Then the generalized version of the Kampé de Fériet function is defined as follows:
For the numerous conditions of convergence for this function, the reader is referred to .
For the purpose of this work, we need to introduce Srivastava’s triple hypergeometric series [, p.44] defined by
where, for convenience,
and abbreviates the array of A parameters with similar interpretations for , , , and so on.
Recently, many authors [12–14] obtained several transformations formulas involving hypergeometric functions as well as their multi-variable analogs by using the so-called beta integral method. The beta function is defined by the following integral representation:
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
The use of a contour of integration in the complex plane provides a very powerful tool in both classical and fractional calculus. The most familiar representation for fractional derivative of order α of is the Riemann-Liouville integral [16–18], that is,
which is valid for , and where the integration is done along a straight line from 0 to z in the ξ-plane. By integrating by parts m times, we obtain
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]).
Definition 2.1 Let be analytic in a simply connected region ℛ. Let be regular and univalent on ℛ and let be an interior point of ℛ. Then if α is not a negative integer, p is not an integer, and z is in , we define the fractional derivative of order α of with respect to by
For non-integer α and p, the functions and in the integrand have two branch lines which begin, respectively, at and , and both pass through the point without crossing the Pochhammer contour at any other point as shown in Figure 1. denotes the principal value of the integrand in (2.3) at the beginning and ending point of the Pochhammer contour which is closed on Riemann surface of the multiple-valued function .
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.
3 The well poised fractional calculus operator
In this section, we recall some of the important properties of the fractional calculus operator introduced by Tremblay  as
We choose to simply list them since the proofs are readily obtainable.
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.
For this work, the most important property of the operator is given by the following relation:
This relation shows, in fact, that the so-called beta integral method consists in a fractional derivative evaluated at the point .
4 Main results
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.
Theorem 4.1 Let and be two nonpositive integers or α be a nonpositive integer and let . Then the following transformation
By making the substitutions and in (3.3), we obtain
Next, we apply the fractional calculus operator on both sides of (4.3) with after operation. We thus have for the l.h.s.:
We obtain for the r.h.s.:
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].
Corollary 4.2 Let and be two nonpositive integers or α be a nonpositive integer and let . Then the following summation formula:
Proof Setting , and in Theorem 4.1 and using twice the Gauss summation formula 
gives the result. □
Theorem 4.3 Let β, c and , and let . Then the following transformation:
and applying the operator on both sides of (4.9), we get for the l.h.s.
and for the r.h.s.
Rewriting (4.11) into the form of (1.2) leads to the desired result. □
Corollary 4.4 Let β, c and . Then the following formula:
Proof Putting in Theorem 4.3, using the Gauss summation formula (4.7) and making elementary simplifications yields the result. □
Corollary 4.5 Let , β, c and . Then the following formula:
Proof Letting and in Theorem 4.3 gives
With the help of the well-known Bailey summation theorem :
the result follows easily after simple calculations. □
Theorem 4.6 Let β, c, λ and . Then the following transformation:
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,
Corollary 4.7 Let β, c, λ and , and . Then the following transformation:
Theorem 4.8 The following transformation:
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].
Corollary 4.9 Let e be a nonpositive integer. Then the following transformation:
Proof Making the following substitutions: , , (4.20) can be written in the form
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.
Theorem 4.10 The following transformation:
if we apply the operator on both sides (4.24), the result follows easily after simple calculations. □
Corollary 4.11 Let α and be two nonpositive integers or a be a nonpositive integer. Then the following transformation:
Proof Putting in Theorem 4.10, we have, after simple manipulations,
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.
Rainville ED: Special Functions. Macmillan Co., New York; 1960.
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Srivastava HM, Karlsson PW: Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester; 1985.
Srivastava HM, Panda R: An integral representation for the product of two Jacobi polynomials. J. Lond. Math. Soc. 1976, 12(2):419–425.
Appell P Mémoire Sci. Math. In Sur les fonctions hypergéométriques de plusieurs variables. Gauthier-Villars, Paris; 1925.
Appell P, Kampé de Fériet J: Fonctions hypergéométriques et hypersphériques: Polynômes d’Hermite. Gauthier-Villars, Paris; 1926.
Slater LJ: Generalized Hypergeometric Functions. Cambridge University Press, London; 1966.
Erdélyi A, Magnus W, Oberhettinger F, Tricomi F: Higher Transcendental Functions, Vols. 1–3. McGraw-Hill, New York; 1953.
Exton H:On Srivastava’s symmetrical triple hypergeometric function . J. Indian Acad. Math. 2003, 25: 17–22.
Srivastava HM: Hypergeometric functions of three variables. Ganita Sandesh 1964, 15: 97–108.
Srivastava HM, Manocha HL: A Treatise on Generating Functions. Ellis Horwood, Chichester; 1984.
Choi J, Rathie AK, Srivastava HM: Certain hypergeometric identities deducible by using the beta integral method. Bull. Korean Math. Soc. 2013, 50: 1673–1681. 10.4134/BKMS.2013.50.5.1673
Krattenthaler C, Rao KS: Automatic generation of hypergeometric identities by the beta integral method. J. Comput. Appl. Math. 2003, 160: 159–173. 10.1016/S0377-0427(03)00629-0
Wei C, Wang X, Li Y: Certain transformations for multiple hypergeometric functions. Adv. Differ. Equ. 2013, 360: 1–13.
Tremblay, R: Une contribution à la théorie de la dérivée fractionnaire. Ph.D. thesis, Laval University, Canada (1974)
Erdélyi A: An integral equation involving Legendre polynomials. SIAM J. Appl. Math. 1964, 12: 15–30. 10.1137/0112002
Liouville J: Mémoire sur le calcul des différentielles à indices quelconques. J. Éc. Polytech. 1832, 13: 71–162.
Riesz M: L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 1949, 81: 1–222. 10.1007/BF02395016
Osler TJ: Fractional derivatives of a composite function. SIAM J. Math. Anal. 1970, 1: 288–293. 10.1137/0501026
Osler TJ: Leibniz rule for the fractional derivatives and an application to infinite series. SIAM J. Appl. Math. 1970, 18: 658–674. 10.1137/0118059
Osler, TJ: Leibniz rule, the chain rule and Taylor’s theorem for fractional derivatives. Ph.D. thesis, New York University (1970)
Osler TJ: Fractional derivatives and Leibniz rule. Am. Math. Mon. 1971, 78: 645–649. 10.2307/2316573
Lavoie J-L, Osler TJ, Tremblay R Lecture Notes in Mathematics. In Fundamental Properties of Fractional Derivatives via Pochhammer Integrals. Springer, Berlin; 1976.
Gaboury S: Some relations involving generalized Hurwitz-Lerch zeta function obtained by means of fractional derivatives with applications to Apostol-type polynomials. Adv. Differ. Equ. 2013., 2013: Article ID 361
Tremblay R, Fugère B-J: The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions. Appl. Math. Comput. 2007, 187: 507–529. 10.1016/j.amc.2006.09.076
Tremblay R, Gaboury S, Fugère B-J: A new Leibniz rule and its integral analogue for fractional derivatives. Integral Transforms Spec. Funct. 2013, 24(2):111–128. 10.1080/10652469.2012.668904
Tremblay R, Gaboury S, Fugère B-J: A new transformation formula for fractional derivatives with applications. Integral Transforms Spec. Funct. 2013, 24(3):172–186. 10.1080/10652469.2012.672323
Tremblay R, Gaboury S, Fugère B-J: Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives. Integral Transforms Spec. Funct. 2013, 24(1):50–64. 10.1080/10652469.2012.665910
Miller KS, Ross B: An Introduction of the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
Vidūnas R: Specialization of Appell’s functions to univariate hypergeometric functions. J. Math. Anal. Appl. 2009, 355: 145–163. 10.1016/j.jmaa.2009.01.047
Bailey WN Cambridge Math. Tracts 32. In Generalized Hypergeometric Series. Cambridge University Press, Cambridge; 1964. Reprinted by Stechert-Hafner, New York
Hasanov A, Turaev M:Decomposition formulas for the double hypergeometric functions and . Appl. Math. Comput. 2007, 187: 195–201. 10.1016/j.amc.2006.08.115
The authors wish to thank the referees for valuable suggestions and comments.
The authors declare that they have no competing interests.
The authors completed the paper together. Both authors read and approved the final manuscript.
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Gaboury, S., Tremblay, R. Remark on certain transformations for multiple hypergeometric functions. Adv Differ Equ 2014, 126 (2014). https://doi.org/10.1186/1687-1847-2014-126
- fractional derivatives
- Appell functions
- Srivastava function
- beta integral
- multiple hypergeometric series