• East Asian mathematical journal
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• 제21권1호
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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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• 제49권4호
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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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• 제29권1호
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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.

• 대한수학회지
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• 제45권2호
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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.

• 대한수학회논문집
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• 제38권4호
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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.

• 한국인터넷방송통신학회논문지
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• 제17권1호
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• pp.59-68
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• 2017

세상에 꽃고 여자도 아름답지만 오일러 공식과 대칭이 가장 아름답다. 샤논은 무선통신과 정보이론이 뿌리가 되는 샤논 정리를 오일러 정리에 기반해 정보와 통신에 응용했고, 오늘날 Smart Phone의 원리다. 그들의 만난 점은 MIMO(multiple input and multiple output) 다중안테나 다이버시티가 $e^{-SNR}$ 이다. 본 논문에서는 세상에서 가장 아름다운 공식 $e^{{\pi}i}+1=0$를 발견한 오일러와 무선통신과 정보통신을 탄생시킨 $C=Blog_2(1+{\frac{S}{N}})$을 발견한 샤논의 공식을 간단히 유도하고 이 두 거장은 샤논 한계(Shannon limit)에서 만남을 증명하고 숨어있는 비밀이 무엇인가를 밝힌다. 또한 대수학코딩이론(Algebraic coding theory)와 삼각함수 속에 숨겨진 비밀은 대칭(symmetric)과 Element-wise Inverse가 됨을 발견한다.

• 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.

• 대한수학회지
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• 제40권1호
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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.

• 호남수학학술지
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• 제26권1호
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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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• 제49권3호
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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.