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On Linear Independence of a Certain Multivariate Infinite Product


Published:20080301
Printed: Mar 2008
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Abstract
Let $q,m,M \ge 2$ be positive integers and
$r_1,r_2,\dots ,r_m$ be positive rationals and
consider the following $M$ multivariate infinite products
\[
F_i = \prod_{j=0}^\infty ( 1+q^{(Mj+i)}r_1+q^{2(Mj+i)}r_2+\dots +
q^{m(Mj+i)}r_m)
\]
for $i=0,1,\dots ,M1$.
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
$\IQ F_0+\IQ F_1 +\dots + \IQ F_{M1} + \IQ$ over $\IQ$ and show that
among these $M$ infinite products, $F_0, F_1,\dots ,F_{M1}$, at least
$\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow
\infty$.