• 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.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.163-186
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• 2018

The objective of this paper is to present 87 recurrence relations for the Jacobi polynomials $P_n^{({\alpha},{\beta})}(x)$. The results presented here most of which are presumably new are obtained with the help of Gauss's fifteen contiguous function relations and some other identities recently recorded in the literature.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.423-436
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• 2018

We aim to establish certain Saigo hypergeometric fractional integral formulas for a finite product of the generalized k-Bessel functions, which are also used to present image formulas of several integral transforms including beta transform, Laplace transform, and Whittaker transform. The results presented here are potentially useful, and, being very general, can yield a large number of special cases, only two of which are explicitly demonstrated.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1673-1681
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• 2013

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.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.205-213
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• 2022

In this paper we give a simple and direct proof of an Euler integral representation for a special class of _q+1F_q,k k-hypergeometric functions for q ≥ 2. The values of certain ₃F_2,k and ₄F_3,k functions at $x=\frac{1}{k}$, some of which can be derived using other methods. We may conclude that for k = 1 the results are reduced to [3].

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.107-110
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• 1996

Usng the Pochhammer symbol $(\lambda)_n$ given by $$ (1.1) (\lambda)_n = {1, if n = 0 {\lambda(\lambda + 1) \cdots (\lambda + n - 1), if n \in N = {1, 2, 3, \ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;\upsilon}^{p:r;u} [\alpha_1, \ldots, \alpha_p : \gamma_1, \ldots, \gamma_r; \lambda_1, \ldots, \lambda_u;_{x,y}][\beta_1, \ldots, \beta_q : \delta_1, \ldots, \delta_s; \mu_1, \ldots, \mu_v; ] = \sum_{l,m = 0}^{\infty} \frac {\prod_{j=1}^{q} (\beta_j)_{l+m} \prod_{j=1}^{s} (\delta_j)_l \prod_{j=1}^{v} (\mu_j)_m)}{\prod_{j=1}^{p} (\alpha_j)_{l+m} \prod_{j=1}^{r} (\gamma_j)_l \prod_{j=1}^{u} (\lambda_j)_m} \frac{l!}{x^l} \frac{m!}{y^m} $$ provided that the double series converges.

• Communications of the Korean Mathematical Society
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• v.30 no.2
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• pp.103-108
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• 2015

Fox [2] presented an interesting identity for $_pF_q$ which is expressed in terms of a finite summation of $_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox [2] found a very interesting and general summation formula for $_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for $_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$_3F_2\[\array{\hspace{110}{\alpha},{\beta},{\gamma};\\{\alpha}-m,\;\frac{1}{2}({\beta}+{\gamma}+i+1);}\;{\frac{1}{2}}\]$$, m being a nonnegative integer and i any integer, can be easily established by suitably specializing the above-mentioned Fox's general identity with, here, the aid of generalizations of Kummer's second summation theorem for $_2F_1(1/2)$ obtained recently by Rakha and Rathie [7]. Several known results are also seen to be certain special cases of our main identities.