Communications of the
Korean Mathematical Society CKMS

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 ${}_{2}F_{1}( frac45 )$- or ${}_{2}F_{1}( frac12 )$-series, or Ramanujan's ${}_{3}F_{2}(1)$-series for the moments of the complete elliptic integral $ ext{{f 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

MSC numbers: Primary 33C20, 33C75