\(p\)-adic invariant summation of some \(p\)-adic functional series (Q2354806)

The authors are interested in \(p\)-adic invariant summation of a class of infinite functional series whose terms contain \(n!\), i.e., \(\sum n! p_k(n;x)x^n\), where \(p_k(n;x)\) are polynomials in \(x\) of degree \(k\), and whose coefficients depend on \(n\). They show that there exist polynomials \(p_k(n;x)\) for any degree \(k\), such that for any \(x\) the corresponding sums are also rational numbers. The summation formula obtained in the paper is a generalization of the one obtained by the first author [Facta Univ., Ser. Math. Inf. 14, 1--10 (1999; Zbl 1048.11016)], when \(x=1\) in \(\sum n! p_k(n;x)x^n\). Some examples are also given to illustrate the results obtained in the paper.