Double series transforms derived from Fourier--Legendre theory
Commun. Korean Math. Soc. 2022 Vol. 37, No. 2, 551-566
Published online March 29, 2022
Printed April 30, 2022
John Maxwell Campbell, Wenchang Chu
York University; University of Salento
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 ${}_{2}F_{1}(\tfrac45 )$- or ${}_{2}F_{1}(\tfrac12 )$-series, or Ramanujan's ${}_{3}F_{2}(1)$-series for the moments of the complete elliptic integral $\text{{\bf K}}$. 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
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