Arithmetic properties of the values some hypergeometric $F$-series

A. Kh. Munos Vaskes

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 519-527

Résumé

Generalized hypergeometric series are of the form \begin{equation} f(z)=\sum_{n=0}^{\infty}\dfrac{(a_1)_n\ldots(a_l)_n}{(b_1)_n\ldots(b_m)_n}z^{n} \end{equation} If $l$ and if the parameters are rational, they are closely related to Siegel's $E$-functions. If $l=m$ and if the parameters are rational, they are $G$-functions. For $l>m$ and if the parameters are rational, they are $F$-series. The arithmetic properties values of generalized hypergeometric series is an actual problem with a long history. We shall only mention Siegel C. L., Shidlovskii A. B., Salikhov V. Kh., Beukers F., Brownawell W. D., Heckman G., Galochkin A. I., Oleinikov V. A., Ivankov P. L., Gorelov V. A., Chirskii V. G., Zudilin W., Matala–Aho T. etc. We consider the so–called $F$-series. Chirskii V.G. proved the infinitу algebraic independence of the corresponding values. Here we obtain lower estimates of polynomials and linear forms in the values of these series and their derivatives in a concrete $p$-adic field.

Détail

Keywords: $F$-series, estimates linear forms and polynomials, $p$-adic numbers.