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http://dx.doi.org/10.4134/CKMS.c210144

DOUBLE SERIES TRANSFORMS DERIVED FROM FOURIER-LEGENDRE THEORY  

Campbell, John Maxwell (Department of Mathematics and Statistics York University)
Chu, Wenchang (Department of Mathematics and Physics University of Salento)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.2, 2022 , pp. 551-566 More about this Journal
Abstract
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.
Keywords
Bivariate hypergeometric series; central binomial coefficient; closed form; Landau's constants; dilogarithm;
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Times Cited By KSCI : 1  (Citation Analysis)
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