Note on the irrationality of certain multivariate \(q\)-functions (Q2495244)

The main result of this paper is the following Theorem: Let \(K\) be either \(\mathbb{Q}\) or an imaginary quadratic number field and \(O_K\) be the ring of integers in \(K\). Let \(q\in O_K\) with \(\mid q\mid >1\). Assume that \(l, m\in \mathbb{N}\) such that \[ m\geq l \quad \text{and} \quad m(l+m)^2>l^2(l(m-1)+m^2) \] or \[ l\geq m \quad \text{and} \quad (l+m)^2>l(m(m-1)+l^2). \] Suppose that \(r_1,\cdots ,r_l\in K\setminus \{ 0\}\) such that \(1+\sum_{n=1}^l q^{-kn}r_n\not= 0\) holds for each \(k\in \mathbb{N}\cup \{ 0\}\). Then at least one among the \(m\) numbers \[ \sum_{i=0}^\infty q^{-mi}\prod_{j=0}^i (1+q^{-(mj+s)}r_1+\cdots +q^{-(mj+s)l}r_l) \quad (s=0,\dots ,m-1) \] is not in \(K\). In addition at least one of the infinite products \[ \prod_{j=0}^\infty (1+q^{-(mj+s)}r_1+\cdots +q^{-(mj+s)l}r_l) \quad (s=0,\dots ,m-1) \] is not in \(K\). The proofs are based on Padé approximations.