  The idea to use classical hypergeometric series and, in particular,
well-poised hypergeometric series in diophantine problems of the values of the
polylogarithms has led to several novelties in number theory and neighbouring
areas of mathematics. Here we present a systematic approach to derive
second-order polynomial recursions for approximations to some values of the
Lerch zeta function, depending on the fixed (but not necessarily real)
parameter $\alpha$ satisfying the condition $\Re(\alpha)<1$. Substituting
$\alpha=0$ into the resulting recurrence equations produces the famous
recursions for rational approximations to $\zeta(2)$, $\zeta(3)$ due to
Ap\'ery, as well as the known recursion for rational approximations to
$\zeta(4)$. Multiple integral representations for solutions of the constructed
recurrences are also given.
