Arithmetic properties of certain polyadic series (Q355243)

The ring of integer polyadic numbers is a direct product of rings \(\mathbb Z_p\) of integer \(p\)-adic numbers. The canonical representation of a polyadic number has the form \(\sum_{n=0}^\infty a_n\cdot n!\), \(0\leq a_n\leq n\). First the author proves the following identity. Theorm 1. For any \(p(x)\in\mathbb Z[x]\) the equality \[ \sum_{n=0}^\infty p(n) \cdot n! = A\cdot\sum_{n=0}^\infty n! + B \] is valid, where \(A,B\in\mathbb Z\) (exact values of \(A\) and \(B\) are presented in the proof). Then Theorem 2 describes arithmetic properties of the considered set of numbers as follows. Theorem 2. The degree of transcendence over \(\mathbb Q\) of the set of polyadic numbers of the form \(\sum_{n=0}^\infty p(n)\cdot n!\) equals 1.