- Research
- Open access
- Published:
Relations between Lauricella’s triple hypergeometric function and Exton’s function
Advances in Difference Equations volume 2013, Article number: 34 (2013)
Abstract
Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function and the Srivastava function by simply splitting Lauricella’s triple hypergeometric function into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al. and also include the relationship between and due to Exton.
MSC: 33C20, 44A45.
1 Introduction and preliminaries
In the usual notation, let ℂ denote the set of complex numbers. For
the generalized hypergeometric series with p numerator parameters and q denominator parameters is defined by (see, for example, [[1], Chapter 4]; see also [[2], p.73]):
where
and is the Pochhammer symbol or the shifted factorial since
which is defined (for ), in terms of the familiar gamma function Γ, by
it being understood conventionally that .
By using the notations as described in [[3], p.33, Equation 1.4(1)] (with ) (see also [[4], p.114, Equation (1)], [[5], Equation (2.1)] and [[6], p.60, Equation 1.7(1)]), Lauricella’s triple hypergeometric function is defined by
Motivated essentially by the works by Lardner [7] and Carlson [8], by simply splitting Lauricella’s triple hypergeometric function into eight parts, very recently Choi et al. [5] presented several relationships between and the Srivastava function (see also [9]). The widely-investigated Srivastava’s triple hypergeometric function , which was introduced over four decades ago by Srivastava [[10], p.428] (see also [5], [[3], p.44, Equation 1.5(14)] and [[6], p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions (see [11], [[12], pp.113-114]) and Srivastava’s three functions , and (see [[13], pp.99-100]; see also [14–16], [[3], p.43] and [[6], pp.60-68]).
Exton’s triple hypergeometric function (see [17]; see also [[3], p.84, Entry 41a and p.101, List 41a]) is defined by
In fact, in 1982 Exton [17] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as and and introduced a set of 20 distinct triple hypergeometric functions to and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions , , a Humbert function and a Humbert function in their kernels. It is not out of place to mention here that Exton’s functions to have been studied a lot until today; see, for example, the works [12, 18–23] and [24]. Moreover, Exton [17] presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature Kummer’s first and second transformations or theorems.
Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al. [25] and also include the relationship between and due to Exton [17].
3 Main transformation formulae
The results to be established here are as follows:
where the coefficients and can be obtained by simply changing n and α into r and , respectively, in Table 1 of and .
Proof For convenience and simplicity, by denoting the left-hand side of (3.1) by S and using the series definition of as given in (1.4), after a little simplification, we have
Using the binomial theorem (see, for example, [[2], p.58]) for the last factor, we get
Using the identity , after a little simplification, we obtain
Now using the following well-known formal manipulation of double series (see [[2], p.56]; for other manipulations, see also [[26], Eq. (1.4)]):
after a little simplification, we have
Using the following formula:
after a little simplification, we get
Using the definition of in (1.1) for the inner series, we obtain
Separating r into even and odd integers, we have
Making use of the following identity:
after a little simplification, we get
Finally, using the known results (2.1) and (2.2), after a little simplification, we easily arrive at the right-hand side of (3.1). This completes the proof of (3.1). □
4 Special cases
In our main formula (3.1), if we take and ±2, after a little simplification, and interpret the respective resulting right-hand sides with the definition of Exton’s triple hypergeometric series given in (1.5), we get the following very interesting relations between and :
The case .
The case .
The case .
The case .
The case .
Remark Clearly, Equation (4.1) is Exton’s result (see [17]) and Equations (4.2) to (4.5) are closely related to it. The other special cases of (3.1) can also be expressed in terms of in a similar manner.
References
Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG I. In Higher Transcendental Functions. McGraw-Hill, New York; 1953.
Rainville ED: Special Functions. Macmillan Co., New York; 1960. Reprinted by Chelsea, New York (1971)
Srivastava HM, Karlsson PW: Multiple Gaussian Hypergeometric Series. Horwood, Chichester; 1985.
Appell P, Kampé de Fériet J: Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite. Gauthier-Villars, Paris; 1926.
Choi J, Hasanov A, Srivastava HM:Relations between Lauricella’s triple hypergeometric function and the Srivastava function . Integral Transforms Spec. Funct. 2012, 23(1):69-82. 10.1080/10652469.2011.596710
Srivastava HM, Manocha HL: A Treatise on Generating Functions. Horwood, Chichester; 1984.
Lardner TJ:Relations between and Bessel functions. SIAM Rev. 1969, 11: 69-72. 10.1137/1011007
Carlson BC:Some extensions of Lardner’s relations between and Bessel functions. SIAM J. Math. Anal. 1970, 1(2):232-242. 10.1137/0501021
Choi J, Kim YS, Hasanov A:Relations between the hypergeometric function of Appell and Kampé de Fériet functions. Miskolc Math. Notes 2011, 12(2):131-148.
Srivastava HM: Generalized Neumann expansions involving hypergeometric functions. Proc. Camb. Philos. Soc. 1967, 63: 425-429. 10.1017/S0305004100041359
Lauricella G: Sulle funzioni ipergeometriche a più variabili. Rend. Circ. Mat. Palermo 1893, 7: 111-158. 10.1007/BF03012437
Kim YS, Choi J, Rathie AK:Remark on two results by Padmanabham for Exton’s triple hypergeometric series . Honam Math. J. 2005, 27(4):603-608.
Srivastava HM: Hypergeometric functions of three variables. Ganita 1964, 15(2):97-108.
Choi J, Hasanov A, Srivastava HM, Turaev M: Integral representations for Srivastava’s triple hypergeometric functions. Taiwan. J. Math. 2011, 15: 2751-2762.
Srivastava HM: Some integrals representing triple hypergeometric functions. Rend. Circ. Mat. Palermo 1967, 16: 99-115. 10.1007/BF02844089
Turaev M:Decomposition formulas for Srivastava’s hypergeometric function on Saran functions. J. Comput. Appl. Math. 2009, 233: 842-846. 10.1016/j.cam.2009.02.050
Exton H: Hypergeometric functions of three variables. J. Indian Acad. Math. 1982, 4: 113-119.
Choi J, Hasanov A, Turaev M: Decomposition formulas and integral representations for some Exton hypergeometric functions. J. Chungcheong Math. Soc. 2011, 24(4):745-758.
Choi J, Hasanov A, Turaev M:Linearly independent solutions for the hypergeometric Exton functions and . Honam Math. J. 2010, 32(2):223-229.
Choi J, Hasanov A, Turaev M:Certain integral representations of Euler type for the Exton function . Honam Math. J. 2010, 32(3):389-397. 10.5831/HMJ.2010.32.3.389
Choi J, Hasanov A, Turaev M:Certain integral representations of Euler type for the Exton function . J. Korean Soc. Math. Edu., Ser. B, Pure Appl. Math. 2010, 17(4):347-354.
Kim YS, Rathie AK:On an extension formulas for the triple hypergeometric due to Exton. Bull. Korean Math. Soc. 2007, 44(4):743-751. 10.4134/BKMS.2007.44.4.743
Kim YS, Rathie AK, Choi J:Another method for Padmanabham’s transformation formula for Exton’s triple hypergeometric series . Commun. Korean Math. Soc. 2009, 24(4):517-521. 10.4134/CKMS.2009.24.4.517
Lee SW, Kim YS: An extension of the triple hypergeometric series by Exton. Honam Math. J. 2009, 31(1):61-71.
Kim YS, Rakha MA, Rathie AK: Generalization of Kummer’s second summation theorem with applications. Comput. Math. Math. Phys. 2010, 50(3):387-402. 10.1134/S0965542510030024
Choi J: Notes on formal manipulations of double series. Commun. Korean Math. Soc. 2003, 18(4):781-789.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This paper was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have equal contributions to each part of this paper. All authors have read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Choi, J., Rathie, A.K. Relations between Lauricella’s triple hypergeometric function and Exton’s function . Adv Differ Equ 2013, 34 (2013). https://doi.org/10.1186/1687-1847-2013-34
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-34