On linear global relations (Q1406412)

Let \[ F(\mu_1,\dots,\mu_m) = \sum\limits_{n=0}^\infty (\mu_1)_n\dots(\mu_m)_n z^n/n!, \] where \(\,(\mu)_0 = 1\), \(\,(\mu)_n = \mu(\mu+1)\dots(\mu+n-1)\), \(\,n\geq 1\,\) and \[ \begin{gathered} f_0(z) = F(\alpha_1,\dots,\alpha_m),\quad\,\, f_1(z) = F(\alpha_1+1,\alpha_2,\dots,\alpha_m),\quad\,\,\dots,\\ f_q(z) = F(\alpha_1+1,\alpha_2+1,\dots,\alpha_{m-1}+1,\alpha_m),\quad\,\, q=m-1.\end{gathered} \] It is assumed that the numbers \(\,\alpha_1,\dots,\alpha_m\,\) and \(z_0\) are the integers from the field \({\mathbb K}\) and there are not less than two integer rational numbers among \(\,\alpha_1,\dots,\alpha_m\). The main result of the paper is the assertion. If the numbers \(\,\alpha_1,\dots,\alpha_m\,\) and \(z_0\) satisfy the above conditions, and \(\,z_0\neq 0\), then between \(\,f_0(z_0),\dots,f_q(z_0)\,\) there are no global linear relations \[ a_0f_0(z_0) + \dots a_qf_q(z_0) = 0, \] where \(\,a_i\in{\mathbb K}\).