Approximants de Padé des $q$-polylogarithmes

Abstract

We solve a Padé-type problem of approximating three specific functions simultaneously by $q$-analogues of polylogarithms, respectively by powers of the logarithm. This problem is intimately related to recent results of the authors and Wadim Zudilin ["Séries hypergéométriques basiques, fonction $q$-zêta et séries d'Eisenstein", J. Inst. Math. Jussieu (to appear)] on the dimension of the vector space generated by $q$-analogues of values of the Riemann zeta function at integers. We also show that our result can be considered as a $q$-analogue of a result of Stéphane Fischler and the second author [J. Math. Pures Appl. {\bf 82} (2003), 1369-1394].