A third-order Apery-like recursion for $ζ(5)$

Abstract

In 1978, Apery has given sequences of rational approximations to $ζ(2)$ and $ζ(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $ζ(4)$ has been discovered. The note contains a possible generalization of the above results for the number $ζ(5)$.