• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.557-573
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• 2021

In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric ₂F₁(x) function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the n-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential are also deduced.

• 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.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.32 no.3
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• pp.677-688
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• 2017

The composition of Jacobi type finite classes of the classical orthogonal polynomials with two generalized Riemann-Liouville fractional derivatives are considered. The outcomes are expressed in terms of generalized Wright function or generalized hypergeometric function. Similar composition formulas are also obtained by considering the generalized Riemann-Liouville and $Erd{\acute{e}}yi-Kober$ fractional integral operators.

• Honam Mathematical Journal
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• v.37 no.1
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• pp.113-126
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• 2015

Recently, Liu and Wang generalized Appell's and Lauricella's hypergeometric functions. Motivated by the work of Liu and Wang, the main object of this paper is to present new generalizations of Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differential formulae and recurrence relations are obtained for these new generalized Appell's and Lauricella's functions.

• The Pure and Applied Mathematics
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• v.30 no.2
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• pp.131-138
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• 2023

The aim of this note is to provide two new and interesting closed-form evaluations of the generalized hypergeometric function ₅F₄ with argument $\frac{1}{256}$. This is achieved by means of separating a generalized hypergeometric function into even and odd components together with the use of two known sums (one each) involving reciprocals of binomial coefficients obtained earlier by Trif and Sprugnoli. In the end, the results are written in terms of two interesting combinatorial identities.

• Nonlinear Functional Analysis and Applications
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• v.26 no.2
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• pp.381-392
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• 2021

During the last several decades, a great variety of fractional kinetic equations involving diverse special functions have been broadly and usefully employed in describing and solving several important problems of physics and astrophysics. In this paper, we aim to find solutions of a type of fractional kinetic equations associated with the (p, q)-extended 𝜏 -hypergeometric function and the (p, q)-extended 𝜏 -confluent hypergeometric function, by mainly using the Laplace transform. It is noted that the main employed techniques for this chosen type of fractional kinetic equations are Laplace transform, Sumudu transform, Laplace and Sumudu transforms, Laplace and Fourier transforms, P_𝛘-transform, and an alternative method.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.131-136
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• 2016

The main object of this paper is to present two general integral formulas whose integrands are the integrand given in the integral formula (3) and a finite product of the generalized Bessel function of the first kind.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.995-1003
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• 2014

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas involving various Bessel functions have been presented. Very recently, Choi and Agarwal derived two generalized integral formulas associated with the Bessel function $J_{\nu}(z)$ of the first kind, which are expressed in terms of the generalized (Wright) hypergeometric functions. In the present sequel to Choi and Agarwal's work, here, in this paper, we establish two new integral formulas involving the generalized Bessel functions, which are also expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

• 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 $_3F_2$.