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:
“In the drawing room [of the Queens palace] hung a Venus and Cupid by Michaelangelo, in which, instead of a bit of drapery, the painter has placed Cupids foot between Venuss thighs. Queen Caroline asked General Guise, an old connoisseur, if it was not a very fine piece? He replied Madam, the painter was a fool, for he has placed the foot where the hand should be.”
—Horace Walpole (17171797)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)