Infinite Transcendence of Values of an Euler-Type Series with a Polyadic Liouville Parameter

Abstract

The famous Euler series

is the subject of research in various directions. It diverges in the field of complex numbers, but can be summed, for example, using the Ramis method, yielding the surprising equality

where \(e,\gamma\) are Euler’s constants. On the other hand, this series converges in all \(p\)-adic number fields and can be proven to be infinitely transcendental (meaning that any nonzero polynomial with integer coefficients does not vanish in an infinite set of \(p\)-adic number fields when this series is substituted into it). A natural generalization of the Euler series are the series of the form

where \((\alpha)_{0}=1,(\alpha)_{n}=\alpha(\alpha+1)\cdots(\alpha+n-1), n\geq 1.\) Previously, the infinite transcendence of the values of this series at algebraic points not equal to 0 with a rational parameter \(\alpha\), the infinite irrationality of the values of this series with an algebraic irrational parameter \(\alpha\), and the infinite irrationality of the values of this series with some transcendental parameter \(\alpha\) were proven. The goal of this paper is to prove the infinite transcendence of the values of this series with some transcendental parameter \(\alpha\). This required a further modification of the Siegel–Shidlovskii method.

We also note that series of the type under consideration are useful in constructing pseudo-random numbers.