Structure of Ap\'{e}ry-like series and monotonicity properties for binomial sums
Bull. Korean Math. Soc. 2017 Vol. 54, No. 1, 225-242
https://doi.org/10.4134/BKMS.b160070
Published online January 31, 2017
Emre Alkan
Ko\c{c} University
Abstract : A family of Ap\'{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.
Keywords : Ap\'{e}ry-like series, transcendental number, central binomial coefficient, monotonicity, competitive cheap talk
MSC numbers : 11B65, 11J82, 05A10
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