• Kyungpook Mathematical Journal
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• v.50 no.4
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• pp.473-482
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• 2010

In the present paper we define a q-extension of the Leibniz rule for q-derivatives via Weyl type q-derivative operator. Expansions and summation formulae for the generalized basic hypergeometric functions of one and more variables are deduced as the applications of the main result.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.129-151
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• 2015

In this paper, we prove an $L^p$ version of Donoho-Stark's uncertainty principle for the inverse of the hypergeometric Fourier transform on $\mathbb{R}^d$. Next, using the ultracontractive properties of the semigroups generated by the Heckman-Opdam Laplacian operator, we obtain an $L^p$ Heisenberg-Pauli-Weyl uncertainty principle for the inverse of the hypergeometric Fourier transform on $\mathbb{R}^d$.

• East Asian mathematical journal
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• v.30 no.3
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• pp.283-291
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• 2014

A remarkably large number of inequalities involving the fractional integral operators have been investigated in the literature by many authors. Very recently, Baleanu et al. [2] gave certain interesting fractional integral inequalities involving the Gauss hypergeometric functions. Using the same fractional integral operator, in this paper, we present some (presumably) new fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Saigo, Erd$\acute{e}$lyi-Kober and Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

• Communications of the Korean Mathematical Society
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• v.30 no.2
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• pp.81-92
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• 2015

In this paper, we aim at establishing a generalized fractional integral version of Gr$\ddot{u}$ss type integral inequality by making use of the Gauss hypergeometric function fractional integral operator. Our main result, being of a very general character, is illustrated to specialize to yield numerous interesting fractional integral inequalities including some known results.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.45-60
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• 2023

In this paper we suggest an enhanced Geraghty-type contractive mapping for examining the existence properties of classical nonlinear operators with or without prior degenerates. The nonlinear operators are proved to exist with the imposition of the Geraghty-type condition in a non-empty closed subset of complete metric spaces. To showcase some efficacies of the Geraghty-type condition, convergent rate and stability are deduced. The results are used to study some asymptotic properties of perturbed integral and hypergeometric operators. The results also extend and generalize some existing Geraghty-type conditions.

• 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.

• Korean Journal of Mathematics
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• v.22 no.4
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• pp.699-709
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• 2014

In the present paper, we investigate starlikeness conditions for q-differential operator by using the concept of quantum calculus in the unit disk.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.441-452
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• 2011

The enormous success of the theory of hypergeometric series in a single variable has stimulated the development of a corresponding theory in two and more variables. A wide variety of investigations in the theory of several variable hypergeometric functions have been essentially motivated by the fact that solutions of many applied problems involving partial differential equations are obtainable with the help of such hypergeometric functions. Here, in this trend, we aim at presenting further decomposition formulas for Saran function $F_E$, which are used to give some integral representations of the function $F_E$. We also present a system of partial differential equations for the Saran function $F_E$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.745-758
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• 2011

Generalizing the Burchnall-Chaundy operator method, the authors are aiming at presenting certain decomposition formulas for the chosen six Exton functions expressed in terms of Appell's functions $F_3$ and $F_4$, Horn's functions $H_3$ and $H_4$, and Gauss's hypergeometric function F. We also give some integral representations for the Exton functions $X_i$ (i = 6, 8, 14) each of whose kernels contains the Horn's function $H_4$.

• Kyungpook Mathematical Journal
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• v.48 no.3
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• pp.379-385
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• 2008

Making use of a linear operator, we introduce certain subclass of meromorphically univalent functions in the punctured unit disk and study its properties including some inclusion results, coefficient and distortion problems. Our result generalize many results known in the literature.