On Linear Independence of a Certain Multivariate Infinite Product

Abstract

AbstractLet q,m,M ≥ 2 be positive integers and r1, r2, … , rm be positive rationals and consider the following M multivariate infinite productsfor 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 ℚF0+ℚF1+· · ·+ℚFM–1+ℚ over ℚ and show that among these M infinite products, F0, F1, … , FM–1, at least ∼ M/m(m + 1) of them are irrational for fixed m and M → ∞.