On defining the product r−k⋅∇lδ

Abstract

Let ρ(s) be a fixed infinitely differentiable function defined on R+=[0,∞) having the properties: (i) ρ(s)≥0, (ii) ρ(s)=0 for s≥1, and (iii) ∫Rmδn(x)dx=1 where δn(x)=cmnmρ(n2r2) and cm is the constant satisfying (iii). We overcome difficulties arising from computing ∇lδn and express this regular sequence by two mutual recursions and use a Java swing program to evaluate corresponding coefficients. Hence, we are able to imply the distributional product r−k⋅∇lδ for k=1,2,… and l=0,1,2,… with the help of Pizetti's formula and the normalization.