A third-order Apéry-like recursion for \(\zeta(5)\) (Q1810251)

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 \([1,2,\dots\,,n]\).