Well-poised generation of Ap\'ery-like recursions

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 $\mathrm{\Re }(\alpha )<1$. 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.