Series representations and simulations of isotropic random fields in the Euclidean space

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

On certain integral formulas involving the product of Bessel function and Jacobi polynomial

In the present paper, we establish some interesting integrals involving the product of Bessel function of the first kind with Jacobi polynomial, which are expressed in terms of Kampe de Feriet and Srivastava and Daoust functions. Some other integrals involving the product of Bessel (sine and cosine) function with ultraspherical polynomial, Gegenbauer polynomial, Tchebicheff polynomial, and Legendre polynomial are also established as special cases of our main results. Further, we derive an interesting connection between Kampe de Feriet and Srivastava and Daoust functions.

Zeros of pseudo-ultraspherical polynomials

The pseudo-ultraspherical polynomial of degree n can be defined by [Formula: see text] where [Formula: see text] is the ultraspherical polynomial. It is known that when λ < -n, the finite set [Formula: see text] is orthogonal on (-∞, ∞) with respect to the weight function (1 + x2)λ-½ and when λ < 1 - n, the polynomial [Formula: see text] has exclusively real and simple zeros. Here, we undertake a deeper study of the zeros of these polynomials including bounds, numbers of real zeros, monotonicity and interlacing properties. Our methods include the Sturm comparison theorem, recurrence relations, and the explicit expression for the polynomials.

Uniform Estimates of Ultraspherical Polynomials of Large Order

In this paper we prove the sharp inequalitywhere is the classical ultraspherical polynomial of degree n and order . This inequality can be refined in [0, ] and [, 1], where denotes the largest zero of .

Proof of Lorch's conjecture on ultraspherical polynomials

We use a Volterra integral equation to derive lower bounds for the local maxima of |un(θ)| = (sin θ)λ|P(λ)n(cos θ)|, where P(λ)n (·) is the nth ultraspherical polynomial with parameter 0 < λ < 1. Moreover, inequalities for the critical points and inequalities between the extrema of un(θ) and un−1(θ) are obtained. The results are applied to show that, for every λ, the maxima of (n+λ)1−λ|un(θ)| form a strictly increasing sequence. This establishes a conjecture of Lorch [12, 13].

On the Monotonic Variation of the Zeros of Ultraspherical Polynomials with the Parameters

We show that increases with λ, for 0< λ < 1, being the kth zero of the ultraspherical polynomial and f(λ) a suitable function of λ. As a consequence, some inequalities for and an estimate for can be obtained.

Some Properties of the q-Hermite Polynomials

Heine [7, p. 93] gave the following representation for the Legendre Polynomial {Pn(x)}∞n=owhere fo,n = 1 andSzegö [7, p. 96] generalized this result to the Ultraspherical Polynomial set ﹛Cnλ(x)﹜∞n=o and obtainedwhere