Little q-legendre polynomials and irrationality of certain Lambert series

Certain q-analogs h(p)(1) of the harmonic series, with p = 1/q an integer g reater than one, were shown to be irrational by Erdos (J. Indiana Math. Soc . 12, 1948, 63-66). In 1991-1992 Peter Borwein (J. Number Theory 37, 1991, 253-259; Proc. Cambridge Philos. Soc. 112, 1992, 141-146) used Pade approxi mation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln(p)(2) of the natural logarithm of 2. Recently Am deberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275-283) used the qEKHA D symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apery's proof of irrationality of zeta (2) and zeta (3). They a lso obtain an upper bound for the measure of irrationality, but better uppe r bounds were earlier given by Bundschuh and Vaananen (Compositio Math. 91, 1994, 175-199) and recently also by Matala-aho and Vaananen (Bull. Austral ian Math. Soc. 58, 1998, 15-31) (for ln(p)(2)). In this paper we show how o ne can obtain rational approximants for h(p)(1) and ln(p)(2) (and many othe r similar quantities) by Pade approximation using little q-Legendre polynom ials and we show that properties of these orthogonal polynomials indeed pro ve the irrationality, with an upper bound of the measure of irrationality w hich is as sharp as the upper bound given by Bundschuh and Vaananen for h(p )(1) and a better upper bound as the one given by Matala-aho and Vaananen f or ln(p)(2).