Euler's factorial series at algebraic integer points

Abstract

We study a linear form in the values of Euler's series F (t) = ∞ n=0 n!t n at algebraic integer points α 1 , . . . , α m ∈ Z K belonging to a number field K. Let v|p be a non-Archimedean valuation of K. Two types of non-vanishing results for the linear form Λare derived, the second of them containing a lower bound for the v-adic absolute value of Λ v . The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series ∞ n=0 n−1 k=0 P (k) t n , where P (x) is a polynomial of degree one.