• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.439-447
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• 2015

Very recently the authors have obtained a very interesting reduction formula for the Srivastava's triple hypergeometric series $F^{(3)}$(x, y, z) by applying the so-called Beta integral method to the Henrici's triple product formula for the hypergeometric series. In this sequel, we also present three more interesting reduction formulas for the function $F^{(3)}$(x, y, z) by using the well known identities due to Bailey and Ramanujan. The results established here are simple, easily derived and (potentially) useful.

• Honam Mathematical Journal
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• v.28 no.2
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• pp.221-224
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• 2006

A Kummer-type transformation formula for the generalized hypergeometric function $_2F_2$ deduced by Exton, rederived in two simple and transparent ways by Miller and generalized by Paris, is again derived by another method.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.682-688
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• 2023

The main object of this paper is to establish twenty definite integrals involving Struve functions and modified Srtuve functions associated with hypergeometric functions, which are presumably new and not present in the scientific literature.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.909-914
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• 2017

The recent research investigates the generalization of Bessel function in different forms as its usefulness in various fields of applied sciences. In this paper, we introduce a new modified form of k-Bessel functions called the generalized modified k-Bessel functions and established some of its properties.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.789-797
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• 2017

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and its associated alternating version whose terms contain a (p, q)-extended Gauss' hypergeometric function. Certain upper bounds for the two series are also given.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.127-142
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• 2018

In the present article, authors obtained certain fractional derivative and integral formulas involving incomplete ${\tau}$-hypergeometric function introduced by Parmar and Saxena [14]. Some interesting special cases and consequences of our main results are also considered.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.1-16
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• 2010

By using the generalized operator method given by Burchnall and Chaundy in 1940, the authors present one-dimensional inverse pairs of symbolic operators. Many operator identities involving these pairs of symbolic operators are rst constructed. By means of these operator identities, 11 decomposition formulas for the generalized hypergeometric $_4F_3$ function are then given. Furthermore, the integral representations associated with generalized hypergeometric functions are also presented.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.363-379
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• 2016

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we introduce here the family of the incomplete generalized ${\tau}$-hypergeometric functions $2{\gamma}_1^{\tau}(z)$ and $2{\Gamma}_1^{\tau}(z)$. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second ${\tau}$-Appell hypergeometric functions ${\Gamma}_2^{{\tau}_1,{\tau}_2}$ and ${\gamma}_2^{{\tau}_1,{\tau}_2}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.187-204
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• 2011

B. C. Carlson [Some extensions of Lardner's relations between $_0F_3$ and Bessel functions, SIAM J. Math. Anal. 1(2) (1970), 232-242] presented several useful relations between Bessel and generalized hypergeometric functions that generalize some earlier results. Here, by simply splitting Srivastava's hypergeometric function $H_B$ into eight parts, we show how some useful and generalized relations between Srivastava's hypergeometric functions $H_B$ and $F^{(3)}$ can be obtained. These main results are shown to be specialized to yield certain relations between functions $_0F_1$, $_1F_1$, $_0F_3$, ${\Psi}_2$, and their products including different combinations with different values of parameters and signs of variables. We also consider some other interesting relations between the Humbert ${\Psi}_2$ function and $Kamp\acute{e}$ de $F\acute{e}riet$ function, and between the product of exponential and Bessel functions with $Kamp\acute{e}$ de $F\acute{e}riet$ functions.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.637-651
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• 2017

Very recently, Mubeen, et al. [6] have obtained fifteen contiguous function relations for k-hypergeometric functions with one parameter by the same technique developed by Gauss. The aim of this paper is to obtain seventy-two new and interesting contiguous function relations for k-hypergeometric functions with two parameters. Obviously, for $k{\rightarrow}1$ we recover the results obtained by Cho, et al. [2] and Rakha, et al. [8].