• 한국수학교육학회지시리즈B:순수및응용수학
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• 제29권2호
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• pp.171-177
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• 2022

The aim of this note is to establish two known sums involving central binomial coefficients via a hypergeometric series approach. As an application, we discover two new closed-form evaluations of generalized hypergeometric function.

• 대한수학회보
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• 제54권1호
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• pp.225-242
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• 2017

A family of $Ap{\acute{e}}ry$-like series involving reciprocals of central binomial coefficients is studied and it is shown that they represent transcendental numbers. The structure of such series is further examined in terms of finite combinations of logarithms and arctangents with arguments and coefficients belonging to a suitable algebraic extension of rationals. Monotonicity of certain quotients of weighted binomial sums which arise in the study of competitive cheap talk models is established with the help of a continuous extension of the discrete model at hand. The monotonic behavior of such quotients turns out to have important applications in game theory.

• 대한수학회논문집
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• 제37권2호
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• pp.551-566
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• 2022

We apply Fourier-Legendre-based integration methods that had been given by Campbell in 2021, to evaluate new rational double hypergeometric sums involving ${\frac{{1}}{\pi}}$. Closed-form evaluations for dilogarithmic expressions are key to our proofs of these results. The single sums obtained from our double series are either inevaluable $_2F_1({\frac{4}{5}})$- or $_2F_1({\frac{1}{2}})$-series, or Ramanujan's ₃F₂(1)-series for the moments of the complete elliptic integral K. Furthermore, we make use of Ramanujan's finite sum identity for the aforementioned ₃F₂(1)-family to construct creative new proofs of Landau's asymptotic formula for the Landau constants.