A Wilf–Zeilberger–Based Solution to the Basel Problem With Applications

Abstract

Wilf [Accelerated series for universal constants, by the WZ method, Discrete Math. Theor. Comput. Sci. 3 (1999) 189–192] applied Zeilberger’s algorithm to obtain an accelerated version of the famous series (cid:80) ∞ k =1 1 /k 2 = π 2 / 6 . However, if we write the Basel series (cid:80) ∞ k =1 1 /k 2 as a 3 F 2 (1) -series, it is not obvious as to how to determine a Wilf–Zeilberger (WZ) pair or a WZ proof certificate that may be used to formulate a proof for evaluating this 3 F 2 (1) -expression. In this article, using the WZ method, we prove a remarkable identity for a 3 F 2 (1) -series with three free parameters that Maple 2020 is not able to evaluate directly, and we apply our WZ proof of this identity to obtain a new proof of the famous formula ζ (2) = π 2 / 6 . By applying partial derivative operators to our WZ-derived 3 F 2 (1) -identity, we obtain an identity involving binomial-harmonic sums that were recently considered by Wang and Chu [Series with harmonic-like numbers and squared binomial coefficients, Rocky Mountain J. Math. , In press], and we succeed in solving some open problems given by Wang and Chu on series involving harmonic-type numbers and squared binomial coefficients. Classification: 11Y60, 33F10.