General Properties
The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution.
For positive integers n, the 2n-th moment of this distribution is
where X is any random variable with this distribution and Cn is the nth Catalan number
so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)
Making the substitution into the defining equation for the moment generating function it can be seen that:
which can be solved (see Abramowitz and Stegun §9.6.18) to yield:
where is the modified Bessel function. Similarly, the characteristic function is given by:
where is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving is zero.)
In the limit of approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.
Read more about this topic: Wigner Semicircle Distribution
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