Transcendence of the values of infinite products in several variables

The aim of this paper is to prove the transcendence of certain infinite products. As applications, we get necessary and sufficient conditions for transcendence of the value of $\Pi_{k=0}^{\infty}(1+a_{k}^{(1)}{z_{1}r^{k}}+\cdot\cdot\cdot+a_{k}^{(m)}{z_{m}r^{k}})$ at appropriate algebraic points, where r ≥ 2 is an integer and {a_n ⁽ⁱ⁾}_{n≥ 0} (1 ≤ i ≤ m) are suitable sequences of algebraic numbers.