Background: I'm trying to compute some harmonic sums using Fourier Legendre expansion. For instance, the following expansion $$\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1) \left(\sum _{k=n+1}^{\infty } \frac{(-1)^{k-1}}{k^2}\right)$$ can be used to compute $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$ (see here). This expansion, as well as the first solution of problem above, are given in "On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums" by Marco Cantarini, Jacopo D’Aurizio (also active contributors to this site).
Problem: Now, as I confront higher weight sums, it turns out F-L expansions for following three functions are urgently needed: $$\frac{\text{Li}_2(x)}{x},\frac{\log ^2(1-x)}{x},\frac{\log (x) \log (1-x)}{x}$$ Since there's no good differentiating Legendre polynomials, so far I haven't figured out how to calculate their F-L expansions based on known results. I'd like you to give some suggestions on it. Thank you!
from Hot Weekly Questions - Mathematics Stack Exchange
MHZ
Post a Comment