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On Linear Independence of a Certain Multivariate Infinite Product
Published online by Cambridge University Press: 20 November 2018
Abstract
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
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 $.
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- Copyright © Canadian Mathematical Society 2008
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