Information entropy of Gegenbauer polynomials of integer parameter

TL;DR

This paper develops a new method using trigonometric and complex analysis techniques to derive the exact analytical expression for the information entropy of Gegenbauer polynomials for any degree and integer parameter, extending previous results.

Contribution

Introduces a novel approach to compute the entropy of Gegenbauer polynomials for arbitrary degree and integer parameters, surpassing previous limitations to specific parameter values.

Findings

Derived the analytical entropy expression for Gegenbauer polynomials with arbitrary degree and integer parameter.

Extended the known entropy formulas beyond the special cases of λ=0,1,2.

Provided a general method applicable to a broad class of polynomials in quantum systems.

Abstract

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 $\lambda\in\mathbb{N}$.