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On Linear Independence of a Certain Multivariate Infinite Product

Published online by Cambridge University Press:  20 November 2018

Stephen Choi  and
Ping Zhou
Stephen Choi
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, V5A 1S6 e-mail: kkchoi@cecm.sfu.ca
Ping Zhou
Affiliation:
Department of Mathematics, Statistics, and Computer Science, St. Francis Xavier University, Antigonish, NS, B2G 2W5 e-mail: pzhou@stfx.ca
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Abstract

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Let $q$,$m$,$M\,\ge \,2$ be positive integers and ${{r}_{1}},\,{{r}_{2}},...,\,{{r}_{m}}$ be positive rationals and consider the following $M$ multivariate infinite products

$${{F}_{i}}\,=\,\prod\limits_{j=0}^{\infty }{(1\,+\,{{q}^{-(Mj+i)}}\,{{r}_{1}}\,+\,{{q}^{-2(Mj+i)}}\,{{r}_{2}}\,+\,\cdot \cdot \cdot +\,{{q}^{-m(Mj+i)}}\,{{r}_{m}})}$$

for $i\,=\,0,\,1,\,.\,.\,.\,,\,M\,-\,1$. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space $\mathbb{Q}{{F}_{0}}\,+\,\mathbb{Q}{{F}_{1}}\,+\cdot \cdot \cdot +\,\mathbb{Q}{{F}_{M-1}}\,+\,\mathbb{Q}$ over ℚ and show that among these $M$ infinite products, ${{F}_{0}}\,+\,{{F}_{1}},...,\,{{F}_{M-1}}$ , at least $\sim \,M/m\left( m+1 \right)$ of them are irrational for fixed $m$ and $M\,\to \,\infty $.

Keywords

11J72
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 32 - 46
Copyright
Copyright © Canadian Mathematical Society 2008

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