Functional estimates for derivatives of the modified Bessel function K0 and related exponential functions

Let K0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function ω~n(x):=(-x)nK0(n)(x)n! for positive argument. The function ω~n plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives ω~n(m) with respect to n can be bounded by O((n+1)m/2) while for small and large arguments x the growth even becomes independent of n. These estimates are based on an integral representation of K0 which involves the function gn(t)=tnn!exp(-t) and its derivatives. The estimates then rely on a subtle analysis of gn and its derivatives which we will also present in this paper.