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
Famous quotes containing the words general and/or properties:
“There are two great rules in life, the one general and the other particular. The first is that every one can in the end get what he wants if he only tries. This is the general rule. The particular rule is that every individual is more or less of an exception to the general rule.”
—Samuel Butler (1835–1902)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)