On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums

In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier-Legendre series expansions, arXiv:1710.03221, 2017), about the interplay between hypergeometric functions and Fourier-Legendre (FL) series expansions. In the section "Hypergeometric series related to pi, pi(2) and the lemniscate constant", through the FL-expansion of [x(1 - x)](mu) (with mu + 1 is an element of 1/4 N) we prove that all the hypergeometric series return rational multiples of 1/pi, 1/pi(2) or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of lov x/root x and related functions, we show that in many cases the hypergeometric F-p+1(p)(..., z) function evaluated at z = +/- 1 can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of In the section "Twisted hypergeometric series" we show that the conversion of some F-p+1(p)(...,+/- 1) values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form Sigma(n >= 0) a(n)b(n) where a(n) is a Stirling number of the first kind and Sigma(n >= 0) b(n)z(n) = F-p+1(p)(...; z).