A third-order Apéry-like recursion for \(\zeta(5)\). (English. Russian original) Zbl 1041.11057

Zudilin, V. V.

Math. Notes 72, No. 5, 733-737 (2002); translation from Mat. Zametki 72, No. 5, 796-800 (2002).

The author constructs two hypergeometric sequences both taking values of the shape \(A_n\zeta(5)+B_n\zeta(3)+C_n\) and thence a four-term (thus third order) linear difference equation with coefficients polynomials in the running index \(n\) and with solutions of the shapes \((q_n\zeta(5)-p_n)\) and \((q_n\zeta(3)-\tilde p_n)\). Here the sequences of rationals \((p_n/q_n)\), respectively \(\tilde p_n/q_n\), converge to \(\zeta(5)\), respectively \(\zeta(3)\), at geometric rate, but not fast enough to prove irrationality – the \(q_n\) are integers, but the \(p_n\) and \(\tilde p_n\) have denominators the seventh power of the lcm \([1,2,\dots\,,n]\).

Reviewer: Alf van der Poorten (Killara)

MSC:

11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11Y35	Analytic computations

Keywords:

hypergeometric sequences; Riemann zeta-function