On Linear Independence of a Certain Multivariate Infinite Product

Choi, Stephen ; Zhou, Ping

Canadian mathematical bulletin, Tome 51 (2008) no. 1, pp. 32-46

Résumé

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 $ .

Détail

DOI : 10.4153/CMB-2008-005-7

Mots-clés : 11J72