• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.677-684
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• 2014

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at generalizing the following transformation formula for the Exton's triple hypergeometric series $X_{12}$ and $X_{17}$: $$(1+2z)^{-b}X_{17}\;\left(a,b,c_3;\;c_1,c_2,2c_3;\;x,{\frac{y}{1+2z}},{\frac{4z}{1+2z}}\right)\\{\hfill{53}}=X_{12}\;\left(a,b;\;c_1,c_2,c_3+{\frac{1}{2}};\;x,y,z^2\right).$$ The results are derived with the help of two general hypergeometric identities for the terminating $_2F_1(2)$ series which were very recently obtained by Kim et al. Four interesting results closely related to the Exton's transformation formula are also chosen, among ten, to be derived as special illustrative cases of our main findings. The results easily obtained in this paper are simple and (potentially) useful.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.65-74
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• 2014

The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-284
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• 2019

The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1951-1967
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• 2017

We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.547-556
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• 2012

A new Hardy-Hilbert type integral inequality for double series with weights can be established by introducing a parameter ${\lambda}$ (with ${\lambda}>1-\frac{2}{pq}$) and a weight function of the form $x^{1-\frac{2}{r}}$ (with $r$ > 1). And the constant factors of new inequalities established are proved to be the best possible. In particular, for case $r$ = 2, a new Hilbert type inequality is obtained. As applications, an equivalent form is considered.

• Bulletin of the Korean Chemical Society
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• v.34 no.12
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• pp.3800-3804
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• 2013

In this study, molecular dynamics simulations of SPC/E (extended simple point charge) model have been carried out in the canonical NVT ensemble over the range of temperatures 300 to 550 K with and without Ewald summation. The quaternion method was used for the rotational motion of the rigid water molecule. Radial distribution functions $g_{OO}(r)$, $g_{OH}(r)$, and $g_{HH}(r)$ and self-diffusion coefficients D for SPC/E water were determined at 300-550 K and compared to experimental data. The temperature dependence on the structural and diffusion properties of SPC/E water was discussed.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.357-385
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• 2014

Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.345-355
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• 2014

The main objective of this paper is to obtain a formula containing eleven interesting results for the reducibility of Kamp$\acute{e}$ de F$\acute{e}$riet function. The results are derived with the help of two general results for the series $_2F_1(2)$ very recently presented by Kim et al. Well known Kummer's second theorem and its contiguous results proved earlier by Rathie and Nagar, and Kim et al. follow special cases of our main findings.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.329-342
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• 2016

Recently, Korobov polynomials have been received a lot of attention, which are discrete analogs of Bernoulli polynomials. In particular, these polynomials are used to derive some interpolation formulas of many variables and a discrete analog of the Euler summation formula. In this paper, we extend these family of polynomials to consider the Korobov polynomials of the fifth kind and of the sixth kind. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

• Journal of Korean Medical classics
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• v.37 no.1
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• pp.41-56
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• 2024

Objectives : The purpose of this paper is to model each Xíng(行) of the Yīnyáng Wǔxíng(陰陽五行) theory as a vector, to interpret the Xiāngshēng Xiāngkè(相生相剋) theory as a vector sum, and argue the objectivity and universal applicability of the Xiāngshēng Xiāngkè(相生相剋) theory. Methods : The five xíngs of the Wǔxíng were modeled and expressed as vectors, and the Xiāngshēng Xiāngkè theories were quantitatively explained by vector summation. Results : We calculated the Wǔxíng vectors using the vector sum formula, and found that the Xíng vectors that received mutual support increased in size by about 62%, and the Xíng vectors that received opposition decreased in size by about 38%. Conclusions : This result could be considered as quantitative interpretation of the contents of the Xiāngshēng Xiāngkè(相生相剋) theory which has mostly been explained qualitatively. The results of this study could hopefully provide ideas to quantify various theories based on the Yinyangwuxing theory such as Korean Medicine and other traditional fields in East Asian culture.