Series acceleration formulas for beta values

Abstract

International audience We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)

A single parameter Hermite-Padé series representations for Apéry's constant

“…10 205k 2 + 250k + 77 64 (2k + 1)!5 , obtained initially by Amdeberhan and Zeilberger (1997), see[8] for more details. Analogously, Pilehrood and Pilehrood (2010)[47] deduced the following formula 30n 2 + 16n + 3 2n 3 (n + 1) 3 Θ n Θ n+1 .…”

“…The methods of Amdeberhan (1996) and Pilehrood and Pilehrood (2010) have exactly the same error rate, which are only shifted by one index value. The reason is that one is derived from the other such that both use the same generation mechanism.…”

“…Convergence parameters Pilehrood (2010). with a convergence rate of r ≈ 3, Pilehrood and Pilehrood (2008) with r ≈ 1.45 and the others with r ≈ 0.6.…”