BRIEF MESSAGE

A certain formula of Liouville

Yu. V. Andrianova^a, V. B. Sherstyukov<sup>ab

a</sup> Lomonosov Moscow State University (Moscow)
^b Moscow Center of Fundamental and Applied Mathematics (Moscow)

Abstract: The classical Liouville formula expressing the multiple integral over a multidimensional pyramid through the integral over a segment is discussed. It is shown how the Liouville formula is related to the special sum, containing successive antiderivatives of the integrand. Specific examples are given to illustrate the general result. Along the way, a compact formula for calculating the power moments of an exponential function is proved.

Keywords: multiple integral, Liouville formula, binomial sum, factorial, subfactorial, Euler number.

UDC: 517.37

Received: 14.02.2024
Accepted: 04.09.2024

DOI: 10.22405/2226-8383-2024-25-3-335-342