• Honam Mathematical Journal
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• v.32 no.3
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• pp.389-397
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• 2010

Exton introduced 20 distinct triple hypergeometric functions whose names are Xi (i = 1,$\ldots$, 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions $_0F_1$, $_1F_1$, a Humbert function $\Psi_2$, a Humbert function $\Phi_2$. The object of this paper is to present 25 (presumably new) integral representations of Euler types for the Exton hypergeometric function $X_5$ among his twenty $X_i$ (i = 1,$\ldots$, 20), whose kernels include the Exton function X5 itself, the Exton function $X_6$, the Horn's functions $H_3$ and $H_4$, and the hypergeometric function F = $_2F_1$.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.347-354
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• 2010

Exton [Hypergeometric functions of three variables, J. Indian Acad. Math. 4 (1982), 113~119] introduced 20 distinct triple hypergeometric functions whose names are $X_i$ (i = 1, ..., 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions $_oF_1$, $_1F_1$, a Humbert function ${\Psi}_2$, a Humbert function ${\Phi}_2$. The object of this paper is to present 16 (presumably new) integral representations of Euler type for the Exton hypergeometric function $X_2$ among his twenty $X_i$ (i = 1, ..., 20), whose kernels include the Exton function $X_2$ itself, the Appell function $F_4$, and the Lauricella function $F_C$.

• The Pure and Applied Mathematics
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• v.25 no.1
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• pp.7-16
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• 2018

The main objective of this paper is to demonstrate how one can obtain very quickly so far unknown Laplace transforms of rather general cases of the generalized hypergeometric function $_3F_3$ by employing generalizations of classical summation theorems for the series $_3F_2$ available in the literature. Several new as well known results obtained earlier by Kim et al. follow special cases of main findings.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.87-105
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• 2013

The aim of the present paper is to study some generating functions of generalized Bateman's and Pasternak's polynomials of two variables.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.127-135
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• 2001

The main object of this paper is to present a transformation formula for a finite series involving $_3F_2$ and some identities associated with the binomial coefficients by making use of the theory of Legendre polynomials $P_{n}$(x) and some summation theorems for hypergeometric functions $_pF_q$. Some integral formulas are also considered.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.169-178
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• 2005

Five results closely related to the well-known Preece's identity obtained earlier by Choi and Rathie will be derived here by using some known hypergeometric identities. In addition to this, the identities obtained earlier by Choi and Rathie have also been written in a compact form.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.63-101
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• 2021

In this research paper, an attempt has been made to provide an interesting extension of the well-known and useful Kummer's second theorem. Several applications have also been given.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.469-475
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• 2005

The authors aim at deriving four generalized summation formulas, which, upon specializing their parameters, give many summation identities including, especially, the four very interesting summation formulas due to Ramanujan. The results are derived with the help of generalized Dixon's theorem obtained earlier by Lavoie, Grondin, Rathie, and Arora.

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.109-136
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• 2012

We aim at presenting certain unexpected functional relations among various hypergeometric functions of one or several variables (for example, see the identities in Corollary 5) by making use of Carlson's method employed in his work (Some extensions of Lardner's relations between $_0F_3$ and Bessel functions, SIAM J. Math. Anal. 1(2)(1970), 232-242).

• 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.