• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.107-116
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• 2012

Here, by using the symbolical method introduced by Burchnall and Chaundy, we aim at constructing certain expansion formulas for the generalized hypergeometric function $_pF_q$. In addition, using our expansion formulas for $_pF_q$, we present formulas of an analytic continuation of the Clausen hypergeometric function $_3F_2$, which are much simpler than an earlier known result. We also give some integral representations for $_3F_2$.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.1-16
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• 2010

By using the generalized operator method given by Burchnall and Chaundy in 1940, the authors present one-dimensional inverse pairs of symbolic operators. Many operator identities involving these pairs of symbolic operators are rst constructed. By means of these operator identities, 11 decomposition formulas for the generalized hypergeometric $_4F_3$ function are then given. Furthermore, the integral representations associated with generalized hypergeometric functions are also presented.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.397-409
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• 2015

This paper considers a class of new convolution integral equations whose kernels involve special functions such as the generalized Mittag-Leffler function and the extended Kummer hypergeometric function. Some basic properties of interconnection with the familiar Riemann-Liouville operators are obtained which are used in fiding the solution of the main convolution integral equation. Several consequences are deduced from the main result by incorporating certain extended forms of hypergeometric functions in our present investigation.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.235-245
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• 2006

Applications of Weyl fractional $q$-integral operator to various generalized basic hypergeometric functions including the basic analogue of Fox's H-function have been investigated in the present paper. Certain interesting special cases have also been deduced.

• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.591-601
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• 2016

Integral transforms and fractional integral formulas involving well-known special functions are interesting in themselves and play important roles in their diverse applications. A large number of integral transforms and fractional integral formulas have been established by many authors. In this paper, we aim at establishing some (presumably) new integral transforms and fractional integral formulas for the generalized hypergeometric type function which has recently been introduced by Luo et al. [9]. Some interesting special cases of our main results are also considered.

• East Asian mathematical journal
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• v.16 no.1
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• pp.81-87
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• 2000

Bose and Mitra obtained certain interesting tansformations of the generalized hypergeometric series by using some known summation formulas and employing suitable contour integrations in complex function theory. The authors aim at providing another transformation of the generalized hypergeometric series by making use of the technique as those of Bose and Mitra and a known summation formula, which Bose and Mitra did not use, for the Gaussian hypergeometric series.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.473-506
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• 2016

This paper continues the study of recursion formulas of multivariable hypergeometric functions. Earlier, in [4], the authors have given the recursion formulas for three variable Lauricella functions, Srivastava's triple hypergeometric functions and k-variable Lauricella functions. Further, in [5], we have obtained recursion formulas for the general triple hypergeometric function. We present here the recursion formulas for Exton's triple hypergeometric functions.

• Honam Mathematical Journal
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• v.37 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.619-630
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• 2018

In this paper, we obtain a (p, q)-extension of the Whittaker function $M_{k,{\mu}}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind ${\Phi}_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

• Honam Mathematical Journal
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• v.41 no.4
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• pp.745-756
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• 2019

In this paper, we establish the log convexity and Turán type inequalities of extended (p, q)-beta functions. Likewise, we present the log-convexity, the monotonicity and Turán type inequalities for extended (p, q)-confluent hypergeometric function by utilizing the inequalities of extended (p, q)-beta functions.