• 충청수학회지
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• 제27권3호
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• pp.363-379
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• 2014

In this paper, introducing various separation axioms on a bi-GTS, it has been observed that such separation axioms actually unify the well-known separation axioms on topological spaces. Several characterizations of such separation properties of a bi-GTS are established in terms of ${\gamma}_{{\mu}_i,{\mu}_j}$-closure operator, generalized cluster sets of functions and graph of functions.

• 호남수학학술지
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• 제43권1호
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• pp.26-34
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• 2021

In this paper two interesting double integrals involving product of two generalized hypergeometric functions have been evaluated in terms of gamma function. The results are derived with the help of known integrals involving hypergeometric functions recorded in the paper of Rathie et al. [6]. We also give several very interesting special cases.

• 대한수학회보
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• 제51권4호
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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.

• 대한수학회지
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• 제48권3호
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• pp.655-667
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• 2011

A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

• 호남수학학술지
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• 제36권3호
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• pp.531-542
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• 2014

Recently, Srivastava et al. [18] introduced the incomplete Pochhammer symbol and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. Here we introduce the incomplete Lauricella function ${\gamma}_D^{(n)}$ and ${\Gamma}_D^{(n)}$ of n variables, and investigate certain properties of the incomplete Lauricella functions, for example, their various integral representations, differential formula and recurrence relation, in a rather systematic manner. Some interesting special cases of our main results are also considered.

• 호남수학학술지
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• 제41권1호
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• pp.163-174
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• 2019

The veritable pursuit of this exegesis is to exhibit integrals affined with the Bessel-Struve kernel function, which are explicitly inscribed in terms of generalized (Wright) hypergeometric function and also the product of generalized (Wright) hypergeometric function with sum of two confluent hypergeometric functions. Somewhat integrals involving exponential functions, modified Bessel functions and Struve functions of order zero and one are also obtained as special cases of our chief results.

• 한국통신학회논문지
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• 제40권7호
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• pp.1217-1233
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• 2015

The paper studies characteristics of the minimum mean-square error symmetric scalar quantizers for the generalized gamma, Bucklew-Gallagher and Hui-Neuhoff probability density functions. Toward this goal, asymptotic formulas for the inner- and outermost thresholds, and distortion are derived herein for nonuniform quantizers for the Bucklew-Gallagher and Hui-Neuhoff densities, parallelling the previous studies for the generalized gamma density, and optimal uniform and nonuniform quantizers are designed numerically and their characteristics tabulated for integer rates up to 20 and 16 bits, respectively, except for the Hui-Neuhoff density. The assessed asymptotic formulas are found consistently more accurate as the rate increases, essentially making their asymptotic convergence to true values numerically acceptable at the studied bit range, except for the Hui-Neuhoff density, in which case they are still consistent and suggestive of convergence. Also investigated is the uniqueness problem of the differentiation method for finding optimal step sizes of uniform quantizers: it is observed that, for the commonly studied densities, the distortion has a unique local minimizer, hence showing that the differentiation method yields the optimal step size, but also observed that it leads to multiple solutions to numerous generalized gamma densities.

• 대한수학회지
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• 제54권2호
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• pp.575-598
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• 2017

This paper introduces and studies a generalization of the classical Struve function of order p given by $$_aS_{p,c}(x):=\sum\limits_{k=0}^{\infty}\frac{(-c)^k}{{\Gamma}(ak+p+\frac{3}{2}){\Gamma}(k+\frac{3}{2})}(\frac{x}{2})^{2k+p+1}$$. Representation formulae are derived for $_aS_{p,c}$. Further the function $_aS_{p,c}$ is shown to be a solution of an (a + 1)-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function $_aS_{p,c}$ are investigated, particulary for the case c = -1. As a consequence, $Tur{\acute{a}}n$-type inequalities are established. For a = 2 and c = -1, dominant and subordinant functions are obtained for the Struve function $_2S_{p,-1}$.

• 대한수학회지
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• 제47권6호
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• pp.1283-1297
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• 2010

In this article, the logarithmically complete monotonicity of some functions such as $\frac{1}{[\Gamma(x+1)]^{1/x}$, $\frac{[\Gamma(x+1)]^{1/x}}{x^\alpha}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and $\frac{[\Gamma(x+\alpha+1)]^{1/(x+\alpha})}{[\Gamma(x+1)^{1/x}}$ for $\alpha{\in}\mathbb{R}$ on ($-1,\infty$) or ($0,\infty$) are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.

• East Asian mathematical journal
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• 제29권5호
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• pp.545-550
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• 2013

A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.