Functional Estimates for Derivatives of the Modified Bessel Function $K_{0}$ and related Exponential Functions

Let $K_{0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tilde{\omega}_{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K_{0}^{\left( n\right) }\left( x\right) }{n!}$ for positive argument. The function $\tilde{\omega}_{n}$ 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 $\tilde{\omega}_{n}^{\left( m\right) }$ with respect to $n$ can be bounded by $O\left( \left( n+1\right) ^{m/2}\right)$ 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 $g_{n}\left( t\right) =\frac{t^{n}}{n!}\exp\left( -t\right)$ and their derivatives. The estimates then rely on a subtle analysis of $g_{n}$ and its derivatives which we will also present in this paper.