• 대한수학회논문집
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• 제21권3호
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• pp.569-575
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• 2006

We aim mainly at presenting two generalizations of the well-known Gauss's second summation theorem and Bailey's formula for the series $_2F_1(1/2)$. An interesting transformation formula for $_pF_q$ is obtained by combining our two main results. Relevant connections of some special cases of our main results with those given here or elsewhere are also pointed out.

• 대한수학회보
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• 제48권1호
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• pp.151-156
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• 2011

In this research paper, motivated by the extension of the Euler type I transformation obtained very recently by Rathie and Paris, the authors aim at presenting the extensions of Euler type II transformation. In addition to this, a natural extension of the classical Saalsch$\ddot{u}$tz's summation theorem for the series $_3F_2$ has been investigated. Two interesting applications of the newly obtained extension of classical Saalsch$\ddot{u}$tz's summation theorem are given.

• 호남수학학술지
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• 제31권2호
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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.

• 대한수학회보
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• 제49권3호
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• pp.621-633
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• 2012

By establishing a new summation formula for the series $_3F_2(\frac{1}{2})$, recently Rathie and Pogany have obtained an interesting result known as Kummer type II transformation for the generalized hypergeometric function $_2F_2$. Here we aim at deriving their result by using a very elementary method and presenting two elegant results for certain terminating series $_3F_2(2)$. Furthermore two interesting applications of our new results are demonstrated.

• East Asian mathematical journal
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• 제17권1호
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• pp.129-133
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• 2001

The aim of this research note is to derive the well known Kummer's second theorem by transforming the integrals which represent some generalized hypergeometric functions. This theorem can also be shown by combining two known Bailey's and Preece's identities for the product of generalized hypergeometric series.

• 대한수학회논문집
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• 제32권4호
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• pp.865-874
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• 2017

In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems have been found interesting applications in obtaining various series identities for ${\Pi}$, ${\Pi}^2$ and ${\frac{1}{\Pi}}$. The aim of this research paper is to provide twelve general formulas for ${\frac{1}{\Pi}}$. On specializing the parameters, a large number of very interesting series identities for ${\frac{1}{\Pi}}$ not previously appeared in the literature have been obtained. Also, several other results for multiples of ${\Pi}$, ${\Pi}^2$, ${\frac{1}{{\Pi}^2}}$, ${\frac{1}{{\Pi}^3}}$ and ${\frac{1}{\sqrt{\Pi}}}$ have been obtained. The results are established with the help of the extensions of classical Gauss's summation theorem available in the literature.

• 호남수학학술지
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• 제37권2호
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• pp.245-252
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• 2015

This paper is in continuation of the paper very recently published [New Laplace transforms of Kummer's confluent hypergeometric functions, Math. Comp. Modelling, 55 (2012), 1068-1071]. In this paper, our main objective is to show one can obtain so far unknown Laplace transforms of three rather general cases of generalized hypergeometric function $_2F_2(x)$ by employing generalized Watson's, Dixon's and Whipple's summation theorems for the series $_3F_2$ obtained earlier in a series of three research papers by Lavoie et al. [5, 6, 7]. The results established in this paper may be useful in theoretical physics, engineering and mathematics.

• East Asian mathematical journal
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• 제25권4호
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• pp.481-486
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• 2009

The first object of this note is to show that a summation formula due to Padmanabham for three-variables hypergeometric function $X_8$ introduced by Exton can be proved in a different (from Padmanabham's and his observation) yet, in a sense, conventional method, which has been employed in obtaining a variety of identities associated with hypergeometric series. The second purpose is to point out that one of two seemingly new hypergeometric identities due to Exton was already recorded and the other one is easily derivable from the first one. A corrected and a little more compact form of a general transform involving hypergeometric functions due to Exton is also given.

• Kyungpook Mathematical Journal
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• 제59권2호
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• pp.293-299
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• 2019

A number of identities associated with the product of generalized hypergeometric series have been investigated. In this paper, we aim to establish an identity involving the product of the generalized hypergeometric series $_2F_2$. We do this using the generalized Kummer-type II transformation due to Rathie and Pogany and another identity due to Bailey. The result presented here, being general, can be reduced to a number of relatively simple identities involving the product of generalized hypergeometric series, some of which are observed to correspond to known ones.

• 대한수학회논문집
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• 제18권3호
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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.