Infinite order Euler operators in spaces of holomorphic functions and Stirling numbers

Abstract

We study infinite order Euler operators in the space $H(\Omega)$ of all functions holomorphic
on an open set $\Omega$ in $\mathbb C^N$, with the topology of uniform convergence on compact sets in $\Omega$. In terms of their characteristic functions, we prove necessary and sufficient conditions for the applicability of these operators to $H(\Omega)$. The special case $\Omega=(\mathbb C\backslash\{0\})^N$ is considered. We study the relationship between the two representations for the Euler operator, in which the Stirling numbers of the first and second kinds play a significant role. It is expressed by means of associated functions, one of which is the sum of the Newton interpolation series. The obtained results imply that each entire function of exponential type $0$ on $\mathbb C^N$ can be expanded into a multidimensional Newton interpolation series. A multidimensional version of the Wigert — Leau theorem is proved. We show that in the space $H(\mathbb C^N)$ of all integer functions in $\mathbb C^N$, each Hadamard type operator in $H(\mathbb C^N)$ is an Euler operator, that is, each continuous linear operator in $H(\mathbb C^N),$ for which each monomial is an eigenvector.