A symmetric extension of Zhao's series for $ζ(5)$

Abstract

We prove a four-parameter extension of Zhao's series for $ζ(5)$. Restricting a Wilf-Zeilberger seed of Au to a hyperplane, we obtain a summand in five parameters with zero sum that is symmetric under all their permutations. Summing it from two different starting points gives two trigonometric closed forms, which we prove by Guillera's periodicity argument. The proof is self-contained, and also gives a companion series for $1/π^2$, which we have not found in the literature. Specializing the parameters at roots of unity, we obtain Borwein-Bradley-type identities whose products depend only on $x^2$, $x^3$, $x^4$ or $x^5$. Their Taylor coefficients are Apéry-like series for odd zeta values and for products of zeta values. In particular, we prove five identities conjectured by Au.