On algebraic independence of some subclasses of hypergeometric functions
The author proves some theorems on the algebraic independence of values at algebraic points of some subclasses of hypergeometric \(E\)-functions that have not been considered earlier, but are quite broad. One of them is as follows: Let \[ F_{\overline {\lambda}, \overline {\nu}} (z)= \sum_{n=0}^\infty {{(\nu_1)_n \cdots (\nu_\ell)_n} \over {(\lambda_1)_n \cdots (\lambda_{t+\ell})_n}} \biggl( {z\over t} \biggr)^{tn}, \] where \((\lambda)_n= \lambda (\lambda+1) \dots (\lambda+n -1)\), \(t\geq 1\), \(\ell\geq 0\), \((t, \ell)\not\in \bigcup^3_{k=0} (6, k)\), and \(\nu_i, \lambda_j\in \mathbb{Q} \setminus \{0, - 1, -2, \dots\}\) for \(i=1, \dots, \ell\) and \(j= 1,\dots, t+\ell\). Suppose that the equation \[ \Biggl\{ \prod_{j=1}^{t+\ell} \biggl(z {d\over dz}+ t(\lambda_j -1) \biggr)- z^t \prod_{j=1}^\ell \biggl(z {d\over dz}+ t\nu_j \biggr) \Biggr\} (y) =0 \tag \(*\) \] is linearly homogeneously irreducible, and that \(\alpha\) is a nonzero algebraic number. Then the numbers \(F^{(i)}_{ \overline {\lambda}, \overline {\nu}} (\alpha)\) \((i=0, 1, \dots, t+\ell- 1)\) are algebraically dependent if and only if the equation \((*)\) is reducible and \(\lambda_{j_0}\in \mathbb{N}\) for some \(j_0\in \{1, \dots, t+\ell\}\). The proof is based on the methods and results of \textit{Yu. V. Nesterenko} [Mat. Zametki 5, 587- 598 (1969; Zbl 0212.394)]\ and \textit{V. Kh. Salikhov} [Acta Arith. 53, No. 5, 453-471 (1990; see above); Mat. Sb. 181, No. 2, 189-211 (1990; Zbl 0712.11044), etc.].
scientific article; zbMATH DE number 796770
scientific article