Information entropy of Gegenbauer polynomials of integer parameter

Abstract

17 pages, 1 figure.-- PACS nrs.: 03.67.−a, 02.30.Gp.-- MSC2000 codes: 30E20, 33B10, 33C45, 33F10, 42C05, 81Q99, 94A17. ; ArXiv pre-print available at: http://arxiv.org/abs/0707.0667 ; MR#: MR2371237 (2009b:33015) ; Zbl#: Zbl 1120.33011 ; The position and momentum information entropies of $D$-dimensional quantum systems with central potentials, such as the isotropic harmonic oscillator and the hydrogen atom, depend on the entropies of the (hyper)spherical harmonics. In turn, these entropies are expressed in terms of the entropies of the Gegenbauer (ultraspherical) polynomials $C_n (\lambda)}(x)$, the parameter $\lambda$ being either an integer or a half-integer number. Up to now, however, the exact analytical expression of the entropy of Gegenbauer polynomials of arbitrary degree $n$ has only been obtained for the particular values of the parameter $\lambda=0,1,2$. Here we present a novel approach to the evaluation of the information entropy of Gegenbauer polynomials, which makes use of trigonometric representations for these polynomials and complex integration techniques. Using this method, we are able to find the analytical expression of the entropy for arbitrary values of both $n$ and $\lambda\in\mathbb{N}$. ; The second author (S. Gandy) gratefully acknowledges the hospitality of the Departamento de Matemáticas of the Universidad Carlos III de Madrid, where this research was carried out, as well as financial support from the European Union Socrates/Erasmus Programme. The work of the first and third authors (J. I. de Vicente and J. Sánchez-Ruiz) was supported by Universidad Carlos III de Madrid, Comunidad Autónoma de Madrid (project no. CCG06-UC3M/EST-0690), and Dirección General de Investigación (MEC) of Spain under grant MTM2006-13000-C03-02. The work of the third author was also supported by the Dirección General de Investigación (MEC) of Spain grant FIS2005-00973, and the Junta de Andalucía research group FQM-0207. ; Publicado