Number Theory

    

On a Volterra Analogue of Grunert'sOperational Formula

Authors: Abdelhay Benmoussa

Let $Vf(x) = int_0^x f(t),dt$ denote the Volterra operator. We derive an explicit expansion for the iterated operator $(xV)^n$ in terms of powers of $V$:$(xV)^n = sum_{k=0}^{n-1} (-1)^k a(n-1,k), x^{,n-k} V^{,n+k},$where $a(n,k)$ are the Bessel coefficients (OEIS A001498). This identity may be viewed as an integral analogue of the classical Grunert's operational formula$(xD)^n = sum_{k=0}^n S(n,k), x^k D^k,$where $S(n,k)$ are the Stirling numbers of the second kind. We also obtain a closed integral representation for $(xV)^n$ and give two applications illustrating the operator identity.

Comments: 7 Pages.

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Submission history

[v1] 2025-08-28 15:40:18
[v2] 2025-09-01 23:18:31
[v3] 2025-09-26 22:58:33
[v4] 2025-10-17 23:36:16

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