Borwein-Bradley-type identities for Ramanujan-type series for $1/π^4$

Source abstract

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for $1/π^4$ and $π^4$ with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on $x^3$, $x^4$, $x^6$ or $x^8$. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's $_5F_4$ sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.