New families of double hypergeometric series for constants involving ${1}/{\pi ^2}$

We apply a Fourier–Legendre-based technique recently introduced by Campbell et al. [J. Math. Anal. Appl. 479 (2019), 90–121], to prove new rational double hypergeometric series formulas for expressions involving $ {1}/{\pi ^2}$, especially the constant $ {\zeta (3)}/{\pi ^2}$, which is of number-theoretic interest. Our techniques, applied in conjunction with Bonnet’s recursion formula, give a powerful tool for evaluating double hypergeometric sums containing products of binomial coefficients, and yield many new transformation formulas for double hypergeometric series. The double series considered may be expressed as single sums involving the moments of elliptic-type integrals which have no known symbolic form.