Arjun K. Rathie

On certain hypergeometric identities deducible by using the beta integral method

The main objective of this paper is to show how one can obtain eleven new and interesting hypergeometric identities in the form of a single result from the old ones by mainly employing the known beta integral method which was recently introduced and used in a systematic manner by Krattenthaler and Rao [6]. The results are derived with the help of a generalization of a well-known hypergeometric transformation formula due to Kummer. Several identities including one obtained earlier by Krattenthaler and Rao [6] follow as special cases of our main results.

A generalization of a formula due to Kummer

The main objective of this investigation is to derive a generalization of Kummer’s formula: which was proved independently by Ramanujan. The results presented here are obtained with the help of a known generalization of one of Kummer’s summation theorems for 2 F 1 , which was given recently by Lavoie et al. [14]. Several interesting identities closely related to Ramanujan’s formula follow as special cases of our main results.

An extension of Saalschütz's summation theorem for the series r+3Fr+2

The aim in this research note is to provide an extension of Saalschützs summation theorem for the series r+3Fr+2(1) when r pairs of numeratorial and denominatorial parameters differ by positive integers. The result is obtained by exploiting a generalization of an Euler-type transformation recently derived by Miller and Paris [Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mountain J Math. 2013;43, to appear].

Extension of a quadratic transformation due to Whipple with an application

The aim of this research is to provide an extension of an interesting and useful quadratic transformation due to Whipple. The result is derived with the help of extension of classical Saalschütz’s summation theorem recently added in the literature. The transformation is further used to obtain a new hypergeometric identity by employing the so-called beta integral method introduced and studied systematically by Krattenthaler and Rao.MSC:33C20, 33C05, 33B20.

On new reduction formulas for the Humbert functions Ψ2, Φ2 and Φ3

ABSTRACT New reduction formulas for the Humbert functions (the confluent Appell functions) , and are given.

A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

In 1812, Gauss obtained fifteen contiguous functions rela- tions. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gausss 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper (16) published in Com- puter & Mathematics with Applications 61 (2011), 620-629. In this paper, first we obtained 15 q-contiguous functions relations due to He- nie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q- contiguous functions relations. These q-contiguous functions relations have wide applications.

Evaluation of certain new class of definite integrals

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkovs work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out.

Some results for terminating F12(2) series

AbstractThe aim of this research paper is to find explicit expressions ofn F12[−n,a;2a+i;2]n andn F12[−n,a2n+i;2]n each for i=0,±1,±2,±3,±4,±5. Known results earlier obtained by Kim et al. and Chu follow special cases of our main findings. The results are derived with the help of generalizations of Gauss second, Kummer and Bailey summation theorems for the series F12 obtained earlier by Lavoie et al.MSC:33C05, 33C20.

Relations between Lauricella’s triple hypergeometric function and Exton’s function

Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function FA(3)(x,y,z) and the Srivastava function F(3)[x,y,z] by simply splitting Lauricella’s triple hypergeometric function FA(3)(x,y,z) into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function FA(3)(x,y,z) and Exton’s function X8 in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating F12(2) series which were very recently obtained by Kim et al. and also include the relationship between FA(3)(x,y,z) and X8 due to Exton.MSC: 33C20, 44A45.Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window and the Srivastava function F ( 3 ) [ x , y , z ] Open image in new window by simply splitting Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window and Exton’s function X 8 Open image in new window in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating F 1 2 ( 2 ) Open image in new window series which were very recently obtained by Kim et al. and also include the relationship between F A ( 3 ) ( x , y , z ) Open image in new window and X 8 Open image in new window due to Exton.

Relations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) and Exton’s function X 8

Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function FA(3)(x,y,z) and the Srivastava function F(3)[x,y,z] by simply splitting Lauricella’s triple hypergeometric function FA(3)(x,y,z) into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function FA(3)(x,y,z) and Exton’s function X8 in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating F12(2) series which were very recently obtained by Kim et al. and also include the relationship between FA(3)(x,y,z) and X8 due to Exton.MSC: 33C20, 44A45.Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window and the Srivastava function F ( 3 ) [ x , y , z ] Open image in new window by simply splitting Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) Open image in new window and Exton’s function X 8 Open image in new window in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating F 1 2 ( 2 ) Open image in new window series which were very recently obtained by Kim et al. and also include the relationship between F A ( 3 ) ( x , y , z ) Open image in new window and X 8 Open image in new window due to Exton.