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International Journal of Mathematics and Mathematical Sciences
Volume 24, Issue 6, Pages 361-369

The product of rk and δ on m

University College of Cape Breton, Mathematics and Computer Science, Nova Scotia, Sydney B1P 6L2, Canada

Received 21 December 1999; Revised 4 January 2000

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 δ2 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 rk and Δδ (Δ denotes the Laplacian) for any positive integer k between 1 and m1 inclusive. Cheng and Li (1991) utilized a net δϵ(x) (similar to the δn(x)) and the normalization procedure of μ(x)x+λ to deduce a commutative neutrix product of rk 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 rk and δ ( is the gradient operator) on m. The nice properties of the δ-sequence are fully shown and used in the proof of our theorem.