• East Asian mathematical journal
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• v.21 no.1
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• pp.77-103
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• 2005

The theory of multiple Gamma functions was studied in about 1900 and has, recently, been revived in the study of determinants of Laplacians. There is a class of mathematical constants involved naturally in the multiple Gamma functions. Here we summarize those mathematical constants associated with the Gamma and multiple Gamma functions and will show how they are involved, if possible.

• Kyungpook Mathematical Journal
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• v.49 no.4
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• pp.731-742
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• 2009

In this paper, by introducing parameters ${\lambda}$, ${\alpha}$and two pairs of conjugate exponents (p, q), (r, s) and applying the improved Euler-Maclaurin's summation formula, we establish a reverse of the slightly sharper Hilbert-type inequality. As applications, the strengthened version and the equivalent form are given.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.99-111
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• 2024

In this paper, a new seven-parameter Mittag-Leffler function of a single complex variable is proposed as a generalization of the standard Mittag-Leffler function, certain generalizations of Mittag-Leffler function, hypergeometric function and confluent hypergeometric function. Certain essential analytic properties are mainly discussed, such as radius of convergence, order, type, differentiation, Mellin-Barnes integral representation and Euler transform in the complex plane. Its relation to Fox-Wright function and H-function is also developed.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.559-573
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• 2008

In this paper, two classes of functions, involving a parameter and the classical Euler gamma function, and two functions, involving the classical Euler gamma function, are verified to be logarithmically completely monotonic in $(-\frac{1}{2},\infty)$ or $(0,\infty)$; some inequalities involving the classical Euler gamma function are deduced and compared with those originating from certain problems of traffic flow, due to J. Wendel and A. Laforgia, and relating to the well known Stirling's formula.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1175-1189
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• 2023

The main purpose of this paper is to give some new identities and properties related to Bernoulli type numbers and polynomials associated with the Bessel function of the first kind. We give symmetric properties of the Bernoulli type numbers and polynomials. Moreover, using generating functions and the Faà di Bruno's formula, we derive some new formulas and relations related to not only these polynomials, but also the Bernoulli numbers and polynomials and the Euler numbers and polynomials.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.17 no.1
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• pp.59-68
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• 2017

The flower and woman are beautiful but Euler's theorem and the symmetry are the best. Shannon applied his theorem to information and communication based on Euler's theorem. His theorem is the root of wireless communication and information theory and the principle of today smart phone. Their meeting point is $e^{-SNR}$ of MIMO(multiple input and multiple output) multiple antenna diversity. In this paper, Euler, who discovered the most beautiful formula($e^{{\pi}i}+1=0$) in the world, briefly guided Shannon's formula ($C=Blog_2(1+{\frac{S}{N}})$) to discover the origin of wireless communication and information communication, and these two masters prove a meeting at the Shannon limit, It reveals something what this secret. And we find that it is symmetry and element-wise inverse are the hidden secret in algebraic coding theory and triangular function.

• East Asian mathematical journal
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• v.29 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.109-128
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• 2003

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP$^{n}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$^{n}$ ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

• Honam Mathematical Journal
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• v.26 no.1
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• pp.133-136
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• 2004

By elementry manipulation of series together with summations of Gauss and $Saalsch\ddot{u}tz$, Exton deduced a new two term relation for the hypergeometric function $_3F_2(1)$. The aim of this paper is to derive Exton's result from Thomae's formula, together with two known integral formulas and the Euler's transformation for $_2F_1$.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.493-506
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• 2009

By using the improved Euler-Maclaurin summation formula and estimating the weight coefficients in this paper, a new dual Hardy-Hilbert's inequality and its reverse form are obtained, which are all with two pairs of conjugate exponents (p, q); (r, s) and a independent parameter ${\lambda}$. In addition, some equivalent forms of the inequalities are considered. We also prove that the constant factors in the new inequalities are all the best possible. As a particular case of our results, we obtain the reverse form of a famous Hardy-Hilbert's inequality.