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