The product of r − k and ∇ δ on ℝ m

In the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluate δ 2 when calculating the transition rates of certain particle interactions and using some products such as ( 1 / x ) ⋅ δ . In 1990, Li and Fisher introduced a computable delta sequence in an m -dimensional space to obtain a noncommutative neutrix product of r − k and Δ δ ( Δ denotes the Laplacian) for any positive integer k between 1 and m − 1 inclusive. Cheng and Li (1991) utilized a net δ ϵ ( x ) (similar to the δ n ( x ) ) and the normalization procedure of μ ( x ) x + λ to deduce a commutative neutrix product of r − k and δ for any positive real number k . The object of this paper is to apply Pizetti's formula and the normalization procedure to derive the product of r − k and ∇ δ ( ∇ is the gradient operator) on ℝ m . The nice properties of the δ -sequence are fully shown and used in the proof of our theorem.