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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {FL}$$\end{document}) series expansions. In the section “Hypergeometric series related toπ,π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ,\pi ^2$$\end{document}and the lemniscate constant”, through the FL-expansion of [x(1-x)]μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[x(1-x)]^\mu $$\end{document} (with μ+1∈14N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu +1\in \frac{1}{4}{\mathbb {N}}$$\end{document}) we prove that all the hypergeometric series ∑n≥0(-1)n(4n+1)p(n)14n2nn3,∑n≥0(4n+1)p(n)14n2nn4,∑n≥0(4n+1)p(n)214n2nn4,∑n≥01p(n)14n2nn3,∑n≥01p(n)14n2nn2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{n\ge 0}\frac{(-1)^n(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\\&\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^4,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^3,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2 \end{aligned}$$\end{document}return rational multiples of 1π,1π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\pi },\frac{1}{\pi ^2}$$\end{document} or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of logxx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\log x}{\sqrt{x}}$$\end{document} and related functions, we show that in many cases the hypergeometric p+1Fp(…,z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom {}_{p+1} F_{p}(\ldots , z)$$\end{document} function evaluated at z=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=\pm 1$$\end{document} can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of ∑n≥01(2n+1)214n2nn2,∑n≥01(2n+1)314n2nn2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\frac{1}{(2n+1)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2,\quad \sum _{n\ge 0}\frac{1}{(2n+1)^3}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) \right] ^2. \end{aligned}$$\end{document}In the section “Twisted hypergeometric series” we show that the conversion of some p+1Fp(…,±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom {}_{p+1} F_{p}(\ldots ,\pm 1)$$\end{document} values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form ∑n≥0anbn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n\ge 0} a_n b_n$$\end{document} where an\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_n$$\end{document} is a Stirling number of the first kind and ∑n≥0bnzn=p+1Fp(…;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n\ge 0}b_n z^n = \phantom {}_{p+1} F_{p}(\ldots ;z)$$\end{document}.