Conditions of mutual transcendence for hypergeometric series (Q1078238)

The author considers the algebraic independence of a set of hypergeometric functions defined by \[ \phi (z)=\sum^{m}_{n=1}\frac{1}{\prod^{m}_{k=1}[\lambda_ k+1,n]}(z/m)^{m\quad n}, \] where \([\lambda,n]=\lambda (\lambda +1)... (\lambda +n-1)\) and \(\lambda_ 1,...,\lambda_ m\) are complex numbers. The following theorem is proved: Let \(\phi\) (z) satisfy the linear differential equation \(Ly=1\), suppose that the natural number \(m\geq 2\) and \(m\lambda_ i-m\lambda_ j\not\in {\mathbb{Z}}\), \(1\leq i<j\leq m\). Then the differential equation \(Ly=0\) is linearly irreducible. This theorem generalizes work of \textit{V. Kh. Salikhov} [Izv. Akad. Nauk SSSR, Ser. Mat. 44, No.1, 176-202 (1980; Zbl 0463.10026)] and it follows that the functions \(\phi (z),\phi^{(1)}(z),...,\phi^{(m-1)}(z)\) are algebraically independent over \({\mathbb{C}}(z)\) as in Salikhov's work.