2010/07/13 by Matthieu Deneufchâtel, Deneufchâtel, Matthieu
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math.CA #math.CO
paper · pdf · doi:10.48550/arxiv.1007.2161
arxiv created 2010/07/13 · openalex publication_date 2010/07/13 · arxiv updated 2010/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we describe a general method for computing Selberg-like integrals based on a formula, due to Kaneko, for Selberg-Jack integrals. The general principle consists in expanding the integrand w.r.t. the Jack basis, which is obtained by a Gram-Schmidt orthogonalization process. The resulting algorithm is not very efficient because of this decomposition. But for special cases, the coefficients admit a closed form. As an example, we study the case of the power-sums since for which the coefficients are obtained by manipulating generating series by means of transformations of alphabets. Furthermore, we prove that the integral is a rational function in the number of variables which allows us to study asymptotics. As an application, we investigate the asymptotic behavior when the integrand involves Jack polynomials and power sums.