Some p-adic differential equations

We investigate various properties of p-adic differential equations which have as a solution an analytic function of the form Fk(x) = ∑ n≥0 n!Pk(n)x , where Pk(n) = n +Ck−1n k−1+· · ·+C0 is a polynomial in n with Ci ∈ Z (in a more general case Ci ∈ Q or Ci ∈ Cp) , and the region of convergence is | x |p< p 1 p−1 . For some special classes of Pk(n), as well as for the general case, the existence of the corresponding linear differential equations of the firstand secondorder for Fk(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 ∑ anx . 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 Fk(x) for some x ∈ Z is considered. Some of these differential equations can be related to p-adic dynamics and p-adic information theory.