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Quantitative linear independence of an infinite product and its derivatives

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Abstract

We establish measures for the rational linear independence of 1 and the values of the product \({\prod_{n>0} \left(1 + z/(q^n + q^{-n})\right)}\) and its derivatives at finitely many rational points, q ≠ 0,±1 being a fixed integer. This is a quantitative improvement upon Bézivin’s very recent result in this journal. In contrast to his procedure, we use the method of Padé approximations of the second kind to get the above-mentioned improvement, some generalizations, and several irrationality measures.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    Peter Bundschuh

  2. Department of Mathematics, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland

    Keijo Väänänen

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  1. Peter Bundschuh
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  2. Keijo Väänänen
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Correspondence to Peter Bundschuh.

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Bundschuh, P., Väänänen, K. Quantitative linear independence of an infinite product and its derivatives. manuscripta math. 129, 423–436 (2009). https://doi.org/10.1007/s00229-009-0262-7

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  • DOI: https://doi.org/10.1007/s00229-009-0262-7

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