The Integral Analogue of Grunert's Formula

TL;DR

The paper addresses the lack of a Grunert-type symbolic expansion for the integral operator $xI$, analogous to the well-known expansion for the Euler operator $xD$. By exploiting the Bessel-number coefficients $a(n,k)$, it derives a closed form $(xI)^n = \sum_{k=0}^{n-1} (-1)^k a(n-1,k) \, x^{n-k} I^{n+k}$ for $n\ge 1$, with $a(n,k)=\frac{(n+k)!}{2^k k!(n-k)!}$. It further provides a Volterra-kernel representation $(xI)^n f(x) = \int_0^x K_n(x,t) f(t) dt$, linking the operator to integral kernels and combinatorial structure. The results yield explicit identities for powers, exponentials, and logarithms, expressed through Beta/Gamma, incomplete gamma, and related special functions, thereby unifying differentiation and integration within a common symbolic framework and enabling direct computation of iterated integrals.

Abstract

We establish a closed symbolic expansion for the operator \(x\mathrm{I}\), where \(\mathrm{I}f(x) = \int_0^x f(t)\,dt\), analogous to the Grunert formula for the Euler operator \(xD\). Specifically, we prove that \[ (x\mathrm{I})^n = \sum_{k=0}^{n-1} (-1)^k\, a(n-1,k)\, x^{\,n-k} \mathrm{I}^{\,n+k}, \] where \(a(n,k)\) are the Bessel numbers (OEIS \seqnum{A001498}). This expansion provides a unified symbolic framework for computing iterated integrals and yields new identities involving classical functions and integer sequences.

Theorems & Definitions (3)

• Theorem 1
• proof
• Corollary 2