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 Double series transforms derived from Fourier–Legendre theory Commun. Korean Math. Soc.Published online December 27, 2021 John Maxwell Campbell and Wenchang Chu York University; University of Salento Abstract : We apply Fourier–Legendre-based integration techniques 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 ${}_{2}F_{1}(\frac{4}{5})$- or ${}_{2}F_{1}(\frac{1}{2})$-series, or Ramanujan's ${}_{3}F_{2}(1)$-series for the moments of the complete elliptic integral $K$. Furthermore, we make use of Ramanujan's finite sum identity for the aforementioned ${}_{3}F_{2}(1)$ 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 ladder MSC numbers : 33C20; 33C75 Full-Text :

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