Little q-Legendre polynomials and irrationality of certain Lambert series

Abstract

Certain q-analogs h p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős (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 approximation 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 Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apery's proof of irrationality of ζ(2) and ζ(3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Vaananen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Vaananen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln p (2)). In this paper we show how one can obtain rational approximants for h p(1) and ln p (2) (and many other similar quantities) by Pade approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which 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 for ln p (2).