Some \(p\)-adic differential equations (Q2751727)

This paper introduces both a general method and explicit examples showing how to compute the inhomogeneous first-order and the homogeneous second-order differential equations where \( \sum^{\infty}_{n=0} P(n) n! x^{n} \) is a solution with \( P \) any polynomial. Then, by specializing \( x\), formulae like \( \sum^{\infty}_{n=0} (-1)^{n} n! (n^{5}+203)=121\) and, by using Volkenborn integration on \( \mathbb Z_{p}\), formulae involving Bernoulli numbers are derived. Applications to ``\(p\)-adic physic'' are also mentioned but without any detail. Computations are mainly formal, \( p\)-adic numbers being only used to make formulae meaningful.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].