Let Cλ n(x), n = 0, 1, . . . , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1−x2)λ−1/2. Denote by xnk(λ), k = 1, . . . , n, the zeros of Cλ n(x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ+1)xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some ...

Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We...

In this work, the coefficients of orthogonal polynomials are obtained in closed form. Our formula works for all classes of orthogonal polynomials whose recurrence relation can be put in the form Rn(x) = x Rn−1(x) − αn−2 Rn−2(x). We show that Chebyshev, Hermite and Laguerre polynomials are all members of the class of orthogonal polynomials with recurrence relations of this form. Our formula unif...

converges. Random matrices with external source were introduced and studied by Brézin and Hikami [7, 8, 9, 10, 11], and P. Zinn-Justin [18, 19]. In what follows, we assume that A hasm distinct eigenvalues a1, . . . , am of multiplicities n1, . . . , nm. We consider m fixed and use multi-index notation ~n = (n1, . . . , nm) and |~n| = n1 + · · ·+ nm. The average characteristic polynomial P~n(x) ...

We introduce and study a deformation of commutative polynomial algebras in even number of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [...

We revisit the radial oscillator from standpoint of a free realization. By using oscillator, namely, creation/annihilation operators harmonic we construct an operator that maps eigenfunctions to those oscillator. As polynomial part this relation, obtain Hermite polynomials Laguerre polynomials.

In this paper, the conformable Laguerre and associated differential equations are solved using Laplace transform. The solution is found to be in exact agreement with that obtained power series. addition some of properties polynomial discussed Rodriguez's Formula generating function proposed proved

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, ...