• The Pure and Applied Mathematics
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• v.12 no.3 s.29
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• pp.223-228
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• 2005

In 2001, Choi, Harsh & Rathie [Some summation formulas for the Appell's function $F_1$. East Asian Math. J. 17 (2001), 233-237] have obtained 11 results for the Appell's function $F_1$ with the help of Gauss's summation theorem and generalized Kummer's summation theorem. We aim at presenting 22 more results for $F_1$ with the help of the generalized Gauss's second summation theorem and generalized Bailey's theorem obtained by Lavoie, Grondin & Rathie [Generalizations of Whipple's theorem on the sum of a $_3F_2$. J. Comput. Appl. Math. 72 (1996), 293-300]. Two interesting (presumably) new special cases of our results for $F_1$ are also explicitly pointed out.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.167-176
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• 2009

The aim of this paper is to obtain three interesting results for reducibility of Kamp$\'{e}$ de $\'{e}$riet function. The results are derived with the help of contiguous Gauss's second summation formulas obtained earlier by Lavoie et al. The results obtained by Bailey, Rathie and Nagar follow special cases of our main findings.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.129-135
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• 2012

The aim of this paper is to establish the well-known and very useful classical Saalsch$\ddot{u}$tz's theorem for the series $_3F_2$(1) by following a different method. In addition to this, two summation formulas closely related to the Saalsch$\ddot{u}$tz's theorem have also been obtained. The results established in this paper are further utilized to show how one can obtain certain known and useful hypergeometric identities for the series $_3F_2$(1) and $_4F_3(1)$ already available in the literature.

• The Pure and Applied Mathematics
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• v.20 no.1
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• pp.37-50
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• 2013

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$: $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

• Communications of the Korean Mathematical Society
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• v.14 no.4
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• pp.843-850
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• 1999

The object of this paper is to give certain classes of pre-sumably new product formulas involving the generalized hypergeo-metric series by modifying the elementary method suggested by Bai-ley.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.11-16
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• 1997

In this paper we will study a p-adic analogue of Kummer's theorem[6],[7], which gives the value at x = -1 of a well-piosed $_2F_1$ hypergeometric series.

• Communications of the Korean Mathematical Society
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• v.18 no.3
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• pp.581-586
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• 2003

The aim of this note is to consider some interesting reducible cases of $H_{A}\;and\;H_{C}$ introduced by Srivastava who actually noticed the existence of three additional complete triple hypergeometric functions $H_{A},\;H_{B},\;and\;H_{C}$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.191-200
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• 2015

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.439-446
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• 2015

The main objective of this paper is to establish certain explicit expressions for the Humbert functions ${\Phi}_2$(a, a + i ; c ; x, -x) and ${\Psi}_2$(a ; c, c + i ; x, -x) for i = 0, ${\pm}1$, ${\pm}2$, ..., ${\pm}5$. Several new and known summation formulas for ${\Phi}_2$ and ${\Psi}_2$ are considered as special cases of our main identities.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.807-819
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• 2023

In 2019, Mathur and Solanki [7, 8] obtained a few transformation formulas for Appell, Horn and the Kampé de Fériet functions. Unfortunately, some of the results are well-known and very old results in literature while others are erroneous. Thus the aim of this note is to provide the results in corrected forms and some of the results have been written in more compact form.