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 $
.