• Theoretical Mathematics and Pedagogical Mathematics
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• v.19 no.2
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• pp.87-102
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• 2012

In this paper, we consider the Fourier-type functionals introduced in [16]. We then establish the analytic Feynman integral for the Fourier-type functionals. Further, we get a series expansion of the analytic Feynman integral for the Fourier-type functional $[{\Delta}^kF]^{\^}$. We conclude by applying our series expansion to several interesting functionals.

• Communications of the Korean Mathematical Society
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• v.14 no.4
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• pp.843-850
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• 1999

The object of this paper is to give certain classes of pre-sumably new product formulas involving the generalized hypergeo-metric series by modifying the elementary method suggested by Bai-ley.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.509-512
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• 2007

We aim at deriving Gauss's second summation theorem for the series $_2F_1(1/2)$ by using Euler's integral representation for $_2F_1$. It seems that this method of proof has not been tried.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.139-145
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• 1996

We show how some interesting results involving series summation and the digamma function are established by means of Riemann-Liouville operator of fractional calculus. We derive the relation $$ \frac{\Gamma(\lambda)}{\Gamma(\nu)} \sum^{\infty}_{n=1}{\frac{\Gamma(\nu+n)}{n\Gamma(\lambda+n)}_{p+2}F_{p+1}(a_1, \cdots, a_{p+1},\lambda + n; x/a)} = \sum^{\infty}_{k=0}{\frac{(a_1)_k \cdots (a_{(p+1)}{(b_1)_k \cdots (b_p)_k K!} (\frac{x}{a})^k [\psi(\lambda + k) - \psi(\lambda - \nu + k)]}, Re(\lambda) > Re(\nu) \geq 0 $$ and explain some special cases.

• East Asian mathematical journal
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• v.17 no.1
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• pp.135-146
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• 2001

Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.35-44
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• 2012

Motivated by the extension of classical Dixon's summation theorem for the series $_3F_2$ given by Lavoie, Grondin, Rathie and Arora, the authors aim at deriving four generalized summation formulas, which, upon specializing their parameters, give many summation identities including, especially, the four very interesting summation formulas due to Ramanujan.

• East Asian mathematical journal
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• v.33 no.5
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• pp.527-531
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• 2017

We aim to prove a known summation formula for the series $_4F_3(1)$ by mainly using a similar method as in [2], which is different from that in [3]. The method of proof here as well as that in [2] is potentially useful in getting some other summation formulas for $_pF_q$.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.87-105
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• 2013

The aim of the present paper is to study some generating functions of generalized Bateman's and Pasternak's polynomials of two variables.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.211-220
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• 2005

In this paper we obtain the transformations of the Ramanujan's tenth order mock theta functions under the modular group generators ${\tau}\;{\rightarrow}\;{\tau}\;+\;1\;and\;{\tau}\;{\rightarrow}\;-1/ {\tau}\;where\;q\;=\;e^{{\pi}it}$.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.97-107
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• 2007

Ramanujan gave six identities involving ${\theta}$-function expansion. We extended these identities by introducing a free parameter. We also give the interpretation of one of the identities using theory of partitions.