The harmonic product of δ(x_1,... ,x_N) and δ(x_1) and two combinatorial identities

摘要

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 δ(x_1, ...,x_n) and δ(x_1) was calculated by Kuribayashi and the author in [LK]. In this paper, the result of [LK] is improved to δ(x_1,... ,x_n) o δ(x_1) = 1/(2πρ)δ(x_1,...,x_n) mod infinitesimals where ρ is a positive infinitesimal. Moreover two combinatorial identities are obtained as byproducts.