• 대한수학회논문집
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• 제25권2호
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• pp.185-191
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• 2010

Srivastava 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. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

• 대한수학회논문집
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• 제22권3호
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• pp.369-371
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• 2007

Very recently, by employing an addition theorem for the con-fluent hypergeometric function, Paris has obtained a Kummer-type trans-formation for a $_2F_2(x)$ hypergeometric function with general parameters in the form of a sum of $_2F_2(-x)$ functions. The aim of this note is to derive his result without using the addition theorem.

• 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.

• 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.

• 대한수학회논문집
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• 제32권2호
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• pp.321-332
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• 2017

In this paper, we generalize the extended Appell's and Lauricella's hypergeometric functions which have recently been introduced by Liu [9] and Khan [7]. Also, we aim at establishing some (presumbly) new integral representations and transforms for the extended generalized Appell's and Lauricella's hypergeometric functions.

• 대한수학회논문집
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• 제15권4호
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• pp.707-714
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• 2000

The authors aim at presenting a presumably new transformation formula involving generalized hypergeometric series by making use of series rearrangement technique which is one of the most effective methods for obtaining generating functions or other identities associated with (especially) the hypergeometric series. They also consider a couple of interesting special cases of their main result.

• 대한수학회논문집
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• 제34권1호
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• pp.169-183
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• 2019

The main aim of this paper is to establish a new class of integrals involving the generalized Galu$Galu{\grave{e}}$-type Struve function with the different type of polynomials such as Jacobi, Legendre, and Hermite. Also, we derive the integral formula involving Legendre, Wright generalized Bessel and generalized Hypergeometric functions. The results obtained here are general in nature and can deduce many known and new integral formulas involving the various type of polynomials.

• 호남수학학술지
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• 제32권3호
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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 = $_2F_1$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제17권4호
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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 $F_C$.

• 대한수학회지
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• 제53권5호
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• pp.1183-1210
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• 2016

During the past four decades or so, due mainly to a wide range of applications from natural sciences to social sciences, the so-called fractional calculus has attracted an enormous attention of a large number of researchers. Many fractional calculus operators, especially, involving various special functions, have been extensively investigated and widely applied. Here, in this paper, in a systematic manner, we aim to establish certain image formulas of various fractional integral operators involving diverse types of generalized hypergeometric functions, which are mainly expressed in terms of Hadamard product. Some interesting special cases of our main results are also considered and relevant connections of some results presented here with those earlier ones are also pointed out.