Linear independence of values of a certain generalization of the exponential function II

Abstract

The arithmetical nature of values at rational points of the hypergeometric series \[ _Q G_R(x) := \sum_{n=0}^\infty \frac{R(1) R(2) \cdots R(n)}{Q(1) Q(2) \cdots Q(n)} x^n \] is studied. $R$ and $Q$ are polynomials with integer coefficients. Using deep results on higher con\-gruences going essentially back to Frobenius, Dedekind, Nagell and Schinzel a measure of $\mathbb{Q}$-linear independence of such values is given. In contrast to former investigations no preseribed factorisations of the polynomials $Q$ and $R$ are necessary. Here, however, congruences to those primes $p$ are used for which $Q$ mod $p$ splits completely into a product of linear factors. To get the measure the fact is used that these primes have a Dirichlet--density. In six applications results, proven in some particular cases by different techniques by Carlson [2], Inkeri [15], Ivankov [17,18], Popken [26] and Bundschuh--Wallisser [1] are derived.