Information entropy of Gegenbauer polynomials of integer parameter

Details Information entropy of Gegenbauer polynomials of integer parameter Information entropy of Gegenbauer polynomials of integer parameter

2007-07-04

arxiv.org/abs/0707.0667v1

J. Phys. A: Math. Theor. 40, 8345-8361 (2007)

Physics

Mathematical Physics

19 pages, 1 Postscript figure

Scientific paper

10.1088/1751-8113/40/29/010

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

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