Let wλ x : 1 − x2 λ−1/2 and Pλ,n x be the ultraspherical polynomials with respect to wλ x . Then, we denote the Stieltjes polynomials with respect to wλ x by Eλ,n 1 x satisfying ∫1 −1 wλ x Pλ,n x Eλ,n 1 x x dx 0, 0 ≤ m < n 1, ∫1 −1 wλ x Pλ,n x Eλ,n 1 x xdx / 0, m n 1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials Eλ,n 1 x and the product Eλ,n 1 ...

Window functions are used to reduce Gibbs’ oscillations resulting from the truncation of a Fourier series and they are employed in a variety of signal processing applications including power spectral estimation, beamforming, and digital filter design. In this dissertation, the application of window functions based on the ultraspherical window is explored. First, two methods for evaluating the c...

The system of Gegenbauer or ultraspherical polynomials $Cn (x);n50,1,.. .% is a classical family of polynomials orthogonal with respect to the weight function vl(x)5(12x ) on the support interval @21,11# . Integral functionals of Gegenbauer polynomials with integrand f (x)@Cn (x)#vl(x), where f (x) is an arbitrary function which does not depend on n or l, are considered in this paper. First, a ...

We prove that many of beta, beta prime, gamma, inverse gamma, Student tand ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, many of ultraspherical and Student t-distributions have free divisibility indicator 1.