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An Ap\'ery-like difference equation for Catalan's constant

Wadim Zudilin
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Abstract

Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we obtain a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it. Similar arguments can be put forward to indicate a second-order difference equation and a new continued fraction for $\zeta(4)=\pi^4/90$, and we announce corresponding results at the end of this paper.

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Wadim Zudilin. 2002-01-04. An Ap\'ery-like difference equation for Catalan's constant. https://arxiv.org/abs/math/0201024

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