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An Apéry-like difference equation for Catalan’s constant. (English) Zbl 1093.11075

Summary: 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 derive a new way of fast calculation of Catalan’s constant as well as a new continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for \(\zeta(4)=\pi^4/90\).

MSC:

11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
39A05 General theory of difference equations
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