Diagonal convergence of the Remainder Pad\'e approximants for the Hurwitz zeta function

The Hurwitz zeta function $\zeta(s, a)$ admits a well-known (divergent) asymptotic expansion in powers of $1/a$ involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Pade approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers $\ge 2$. The latter can be viewed, in a certain sense, as analogues of Apery's celebrated sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$.