• Communications of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.885-898
• /
• 2017

The Bailey lemma has been a powerful tool in the discovery of identities of Rogers-Ramanujan type and also ordinary and basic hyper-geometric series identities. The mechanism of Bailey lemma has also led to the concepts of Bailey pair and Bailey chain. In the present work certain new WP-Bailey pairs have been established. We also have deduced a number of basic hypergeometric series identities as an application of new WP-Bailey pairs.

• Communications of the Korean Mathematical Society
• /
• v.18 no.4
• /
• pp.781-789
• /
• 2003

Formal manipulations of double series are useful in getting some other identities from given ones and evaluating certain summations, involving double series. The main object of this note is to summarize rather useful double series manipulations scattered in the literature and give their generalized formulas, for convenience and easier reference in their future use. An application of such manipulations to an evaluation for Euler sums (in itself, interesting), among other things, will also be presented to show usefulness of such manipulative techniques.

• Communications of the Korean Mathematical Society
• /
• v.12 no.1
• /
• pp.11-16
• /
• 1997

In this paper we will study a p-adic analogue of Kummer's theorem[6],[7], which gives the value at x = -1 of a well-piosed $_2F_1$ hypergeometric series.

• Communications of the Korean Mathematical Society
• /
• v.14 no.1
• /
• pp.57-68
• /
• 1999

In this short note, we construct another form of Stirling`s asymptotic series by new form of Carlitz`s q-Bernoulli numbers.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.1
• /
• pp.107-114
• /
• 2018

In this paper, we give some identities for the higher-order Euler functions arising from the Fourier series of them. In addition, we investigate some formulae related to Bernoulli functions which are derived from our identities.

• Honam Mathematical Journal
• /
• v.35 no.3
• /
• pp.507-513
• /
• 2013

In this paper, we compute transformation formulae for generalized non-holomorphic Eisenstein series.

• Honam Mathematical Journal
• /
• v.36 no.4
• /
• pp.895-899
• /
• 2014

In this paper, we consider generalized two variable Eisenstein series. We give analytic continuation and prove modular transformation formulae for them.

• Proceedings of the Korean Statistical Society Conference
• /
• 2002.05a
• /
• pp.49-54
• /
• 2002

For arbitrary random variables {$X_{n},n{\geq}1$}, the order of growth of the series. $S_{n}\;=\;{\sum}_{j=1}^n\;X_{j}$ is studied in this paper. More specifically, when the series S_{n}$ diverges almost surely, the strong law of large numbers $S_{n}/g_{n}^{-1}$($A_{n}{\psi}(A_{n}))\;{\rightarrow}\;0$ a.s. is constructed by extending the results of Petrov (1973). On the other hand, if the series $S_{n}$ converges almost surely to a random variable S, then the tail series $T_{n}\;=\;S\;-\;S_{n-1}\;=\;{\sum}_{j=n}^{\infty}\;X_{j}$ is a well-defined sequence of random variables and converges to 0 almost surely. For the almost surely convergent series $S_{n}$, a tail series strong law of large numbers $T_{n}/g_{n}^{-1}(B_{n}{\psi}^{\ast}(B_{n}^{-1}))\;{\rightarrow}\;0$ a.s., which generalizes the result of Klesov (1984), is also established by investigating the duality between the limiting behavior of partial sums and that of tail series. In particular, an example is provided showing that the current work can prevail despite the fact that previous tail series strong law of large numbers does not work.

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

• Journal of Environmental Science International
• /
• v.20 no.11
• /
• pp.1425-1435
• /
• 2011

Time-series change in Gyeongpo beach shoreline was illustrated using DGPS(Differential Global Positioning System, resolution < 0.6m) observation from April, 2009 to April, 2010. The shoreline was subdivided into 12 areas, and westward and eastward movement of shoreline position at each area was calculated. In general, the shoreline moved toward sea during summer, and it moved toward land during winter. The southern and northern part of the shoreline had different pattern in time-series. The shoreline in the southern part moved toward sea during summer and moved toward land during winter, but time-series pattern of the shoreline in the northern part was more complicated than that in the southern part. Pattern of time-series change in the northern part was made up of three different types; the first is that the shoreline moves continuously toward land, and the second thing is that the shoreline's movement is the opposite to the southern part, and the third thing is that the shoreline maintains a state of equilibrium without any great fluctuation. The total length of the shoreline was the largest during winter and the smallest during summer. In general, time-series change in the shoreline had positive(+) relationship with sea surface pressure and wind speed.