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A third-order Apéry-like recursion for $$\zeta(5)$$. (English. Russian original) Zbl 1041.11057
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 $$[1,2,\dots\,,n]$$.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11Y35 Analytic computations
##### Keywords:
hypergeometric sequences; Riemann zeta-function
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