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.