On defining the product \(r^{-k} \cdot \nabla^{l} \delta\) (Q1774705)

Summary: Let \(\rho(s)\) be a fixed infinitely differentiable function defined on \({\mathbb R}^{+}=[0,\infty)\) having the properties: (i) \(\rho(s)\geq 0\), (ii) \(\rho(s)=0\) for \(s \geq 1\), and (iii) \(\int_{{\mathbb R}^m}\delta_n(x)dx = 1\) where \(\delta_n(x) = c_m n^m \rho(n^2 r^2)\) and \(c_m\) is the constant satisfying (iii). We overcome difficulties arising from computing \(\nabla^{l} \delta_n\) and express this regular sequence by two mutual recursions and use a Java swing program to evaluate the corresponding coefficients. Hence, we are able to define the distributional product \(r^{-k}\cdot\nabla^l\delta\) for \(k = 1, 2,\ldots\) and \(l = 0,1,2,\dotsc\) with the help of Pizetti's formula and the normalization.