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On Linear Independence of a Certain Multivariate Infinite Product

  Published:2008-03-01
 Printed: Mar 2008
  • Stephen Choi
  • Ping Zhou
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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 $i=0,1,\dots ,M-1$. 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 $\IQ F_0+\IQ F_1 +\dots + \IQ F_{M-1} + \IQ$ over $\IQ$ and show that among these $M$ infinite products, $F_0, F_1,\dots ,F_{M-1}$, at least $\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow \infty$.
MSC Classifications: 11J72 show english descriptions Irrationality; linear independence over a field 11J72 - Irrationality; linear independence over a field
 

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