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

Marco Cantarini; Jacopo D'Aurizio

paper

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

arXiv:1806.08411

Abstract

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ( $FL$ ) series expansions. In the section "Hypergeometric series related to $π, π^{2}$ and the lemniscate constant", through the FL-expansion of $[x (1 - x)]^{μ}$ (with $μ + 1 \in \frac{1}{4} N$ ) we prove that all the hypergeometric series $n \geq 0 \sum \frac{( - 1 ) ^{n} ( 4 n + 1 )}{p ( n )} [\frac{1}{4 ^{n}} (n 2 n)]^{3}, n \geq 0 \sum \frac{( 4 n + 1 )}{p ( n )} [\frac{1}{4 ^{n}} (n 2 n)]^{4},$ $n \geq 0 \sum \frac{( 4 n + 1 )}{p ( n ) ^{2}} [\frac{1}{4 ^{n}} (n 2 n)]^{4}, n \geq 0 \sum \frac{1}{p ( n )} [\frac{1}{4 ^{n}} (n 2 n)]^{3}, n \geq 0 \sum \frac{1}{p ( n )} [\frac{1}{4 ^{n}} (n 2 n)]^{2}$ return rational multiples of $\frac{1}{π}, \frac{1}{π ^{2}}$ or the lemniscate constant, as soon as $p (x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{l o g x}{x}$ and related functions, we show that in many cases the hypergeometric $_{p + 1} F_{p} (\dots, z)$ function evaluated at $z = \pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $n \geq 0 \sum \frac{1}{( 2 n + 1 ) ^{2}} [\frac{1}{4 ^{n}} (n 2 n)]^{2}, n \geq 0 \sum \frac{1}{( 2 n + 1 ) ^{3}} [\frac{1}{4 ^{n}} (n 2 n)]^{2} .$ In the section "Twisted hypergeometric series" we show that the conversion of some $_{p + 1} F_{p} (\dots, \pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n \geq 0} a_{n} b_{n}$ where $a_{n}$ is a Stirling number of the first kind and $\sum_{n \geq 0} b_{n} z^{n} =_{p + 1} F_{p} (\dots; z)$ .