Using hypergeometric series, simultaneous approximations for polylogarithms are proposed of the form and where is a polynomial in and and are sums of polynomials in and . By analytic continuation, this gives simultaneous approximations to and in which case Apéry-like recurrence relations of order 3 for and , and hence also for and are obtained.
Two generalizations are given. The first is also including , giving approximations for to and , and as before, recurrence relations for the , , , and . The second generalization introduces well-poised hypergeometric series, which leads for to simultaneous approximations to the numbers and .