Little qLegendre Polynomials and Irrationality of Certain Lambert Series
 Walter Van Assche
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Certain qanalogs 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 Padé approximation and complex analysis to prove the irrationality of these qharmonic series and of qanalogs 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 qWZ pairs that provide a proof of irrationality similar to Apéry'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 Väänänen (Compositio Math. 91, 1994, 175–199) and recently also by Matalaaho and Väänänen (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 Padé approximation using little qLegendre 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 Väänänen for h _{p}(1) and a better upper bound as the one given by Matalaaho and Väänänen for ln_{ p }(2).
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 Title
 Little qLegendre Polynomials and Irrationality of Certain Lambert Series
 Journal

The Ramanujan Journal
Volume 5, Issue 3 , pp 295310
 Cover Date
 20010901
 DOI
 10.1023/A:1012930828917
 Print ISSN
 13824090
 Online ISSN
 15729303
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 little qLegendre polynomials
 irrationality
 measure of irrationality
 Industry Sectors
 Authors

 Walter Van Assche ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B3001, Leuven, Belgium