The harmonic product of $\delta (x_1,...,x_n)$ and $\delta (x_1)$ and two combinatorial identities

Abstract

In the framework of nonstandard analysis, Bang-He Li and the author defined the product of any two distributions on $R^n$ via their harmonic representations. The product of $\delta (x_1,...,x_n)$ and $\delta (x_1)$ was calculated by Kuribayashi and the author in [LK]. In this paper, the result of [LK] is improved to $$\delta (x_1,...,x_n)\circ \delta (x_1)=\frac{1}{2\pi \rho }\delta (x_1,...,x_n)modinfinitesimals$$ where $\rho$ is a positive infinitesimal. Moreover two combinatorial identities are obtained as byproducts.

Latex. To appear in Hiroshima Math. J