Some p-adic differential equations

Abstract

We investigate various properties of p-adic differential equations which have as a solution an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k (n) x^n$, where $P_k (n) = n^k + C_{k-1} n^{k-1} + ...+ C_0$ is a polynomial in n with $C_i\in Z$ (in a more general case $C_i\in Q$ or $C_i\in C_p$). For some special classes of $P_k (n)$, as well as for the general case, the existence of the corresponding linear differential equations of the first- and second-order for $F_k (x)$, is shown. In some cases such equations are constructed. For the second-order differential equations there is no other analytic solution of the form $\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous first-order differential equation exists one can construct infinitely many inhomogeneous second-order equations with the same analytic solution. Relation to some rational sums with the Bernoulli numbers and to $F_k (x)$ for some $x\in Z$ is considered. Some of these differential equations can be related to p-adic dynamics and p-adic information theory.