Apéry-like Difference Equation for Catalan’s Constant

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 ζ(4) = π/90, and we announce corresponding results at the end of this paper. 0. Introduction One of the most crucial and quite mysterious ingredients in Apéry’s proof [Ap], [Po] of the irrationality of ζ(2) and ζ(3) is the existence of the difference equations (n+ 1)un+1 − (11n + 11n+ 3)un − nun−1 = 0, u′0 = 1, u ′ 1 = 3, v ′ 0 = 0, v ′ 1 = 5, (1) and (n+ 1)un+1 − (2n+ 1)(17n + 17n+ 5)un + nun = 0, u 0 = 1, u ′′ 1 = 5, v ′′ 0 = 0, v ′′ 1 = 6, with the following properties of their solutions: lim n→∞ v n un = ζ(2), lim n→∞ v n u′′ n = ζ(3). Unexpected inclusions un, D 2 nv ′ n ∈ Z and u n, D nv n ∈ Z, where Dn denotes the least common multiple of the numbers 1, 2, . . . , n (and D0 = 1 for completeness), together 1991Mathematics Subject Classification. Primary 39A05, 11B37, 11J70; Secondary 33C20, 33C60, 11B65, 11M06.