On defining the product r-k⋅∇lδ

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) Rm δn(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.