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

Let K 0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function ω ˜ n ( x ) : = ( − x ) n K 0 ( 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 K 0 which involves the function g n ( t ) = t n n ! exp ( − t ) and its derivatives. The estimates then rely on a subtle analysis of g n and its derivatives which we will also present in this paper.