• Korean Journal of Mathematics
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• v.27 no.1
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• pp.221-267
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• 2019

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and $^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, and its dual $^t{\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on W-invariant $L^p_{\mathcal{A}k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.149-157
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• 2012

By means of fractional calculus techniques, we find explicit solutions of confluent hypergeometric equations. We use the N-fractional calculus operator $N^{\mu}$ method to derive the solutions of these equations.

• 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.

• 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.34 no.2
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• pp.507-522
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• 2019

The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of beta function recently defined by Shadab et al. [19]. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

• Nonlinear Functional Analysis and Applications
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• v.28 no.1
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• pp.81-93
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• 2023

By utilizing a certain Libera integral operator considered on analytic multivalent functions in the unit disk U. Using the hypergeometric function and the Libera integral operator, we included a new convolution operator that expands on some previously specified operators in U, which broadens the scope of certain previously specified operators. We introduced and investigated the properties of new subclasses of functions f (z) ∈ A_p using this operator.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1239-1248
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• 2018

Recently, many authors have obtained several hypergeometric identities involving hypergeometric functions of one and multi-variables such as the Appell's functions and Horn's functions. In this paper, we obtain several new transformations suitably by applying the fractional calculus operator to these hypergeometric identities, which was introduced recently by Tremblay.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.433-443
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• 2019

In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.147-159
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• 2016

In this paper, subordination results of analytic function $f{\in}{\mathcal{A}}_p$ involving linear operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are obtained. By applying the differential subordination method, results are derived under some sufficient subordination conditions. On using some hypergeometric identities, corollaries of the main results are derived. Furthermore, convolution preserving properties for a class of multivalent analytic function associated with the operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are investigated.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.235-248
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• 2016

The purpose of this paper is to introduce sufficient conditions for (Gaussian) hypergeometric functions to be in various subclasses of analytic functions. Also, we investigate several mapping properties involving these subclasses.