• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.569-576
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• 2004

In 1965, Bhatt and Pandey obtained the Watson's theorem for double series by using Dioxon's theorem on the sum of a $_3F_2$. The aim of this paper is to derive twenty three results for double series closely related to the Watson's theorem for double series obtained by Bhatt and Pandey. The results are derived with the help of twenty three summation formulas closely related to the Dison's theorem on the sum of a $_3F_2$ obtained in earlier work by Lavoie, Grondin, Rathie and Arora.

• East Asian mathematical journal
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• v.39 no.3
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• pp.299-303
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• 2023

The aim of this note is to provide three derivations of the well-known Gregory-Leibniz series for π via a hypergeometric series approach.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.395-403
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• 2005

The authors aim mainly at giving fifteen three-term contiguous relations for the basic hypergeometric series $series\;_2{\phi}_1$ corresponding to Gauss's contiguous relations for the hypergeometric series $series\;_2F_1$ given in Rainville([6], p.71). They also apply them to obtain two summation formulas closely related to a known q-analogue of Kummer's theorem.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.185-191
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• 2010

Srivastava noticed the existence of three additional complete triple hypergeometric functions $H_A$, $H_B$ and $H_C$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

• Communications for Statistical Applications and Methods
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• v.16 no.6
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• pp.1013-1022
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• 2009

We in this paper study the almost sure convergence for asymptotically almost negatively associated(AANA) random variable sequences and obtain some new results which extend and improve the result of Jamison et al. (1965) and Marcinkiewicz-Zygumnd strong law types in the form given by Baum and Katz (1965), three-series theorem.

• The Mathematical Education
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• v.29 no.1
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• pp.55-61
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• 1990

There are three objectives of this paper. First, we present an elegant and simple generalization of Abel's theorem (i .e. tile Abel summability (on the unit disk of the euclidean plane) is regular). Second, we consider the definition of Abel summability through lim (equation omitted) which immediately has clear connexctions with CeSARO summability and Cesaro sums (equation omitted). This approach examplifies some simple aspects of so-called Tauberian theorems of divergent series. Third, we present the applications of the previous results to find the limits of transition probabilities of homogeneous Marker chain. Finally, we explain why the original Abel's theorem which looks obvious is difficult to be proved, and can not be proved analytically.

• Honam Mathematical Journal
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• v.37 no.2
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• pp.245-252
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• 2015

This paper is in continuation of the paper very recently published [New Laplace transforms of Kummer's confluent hypergeometric functions, Math. Comp. Modelling, 55 (2012), 1068-1071]. In this paper, our main objective is to show one can obtain so far unknown Laplace transforms of three rather general cases of generalized hypergeometric function $_2F_2(x)$ by employing generalized Watson's, Dixon's and Whipple's summation theorems for the series $_3F_2$ obtained earlier in a series of three research papers by Lavoie et al. [5, 6, 7]. The results established in this paper may be useful in theoretical physics, engineering and mathematics.

• Communications of the Korean Mathematical Society
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• v.18 no.3
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• pp.581-586
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• 2003

The aim of this note is to consider some interesting reducible cases of $H_{A}\;and\;H_{C}$ introduced by Srivastava who actually noticed the existence of three additional complete triple hypergeometric functions $H_{A},\;H_{B},\;and\;H_{C}$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables.

• East Asian mathematical journal
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• v.25 no.4
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• pp.481-486
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• 2009

The first object of this note is to show that a summation formula due to Padmanabham for three-variables hypergeometric function $X_8$ introduced by Exton can be proved in a different (from Padmanabham's and his observation) yet, in a sense, conventional method, which has been employed in obtaining a variety of identities associated with hypergeometric series. The second purpose is to point out that one of two seemingly new hypergeometric identities due to Exton was already recorded and the other one is easily derivable from the first one. A corrected and a little more compact form of a general transform involving hypergeometric functions due to Exton is also given.