Wigner semicircle distribution
so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)
The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z.
The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals.
The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0):
The characteristic function (CF) of the PDF is related to the beta distribution as shown below
The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are:
Skew and Kurtosis can also be simplified in terms of Bessel functions.
Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining