In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

ABSTRACT: Knop and Sahi introduced a family of non-homogeneous and nonsymmetric polynomials, Gα(x; q, t), indexed by compositions. An explicit formula for the bivariate Knop-Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macd...

formulae expressing explicitly the coefficients of an expansion of double jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. extension to expansion of triple jacobi polynomials is given. the results for the special cases of double and triple ultraspher...

We construct Ultraspherical-weighted orthogonal polynomials C (λ,γ) n,r (u, v, w), λ > − 2 , γ > −1, on the triangular domain T, where 2λ + γ = 1. We show C (λ,γ) n,r (u, v, w), r = 0, 1, . . . , n; n ≥ 0 form an orthogonal system over the triangular domain T with respect to the Ultraspherical weight function. Mathematics Subject Classification: 33C45, 42C05, 33C70

Abstract. We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q = 1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of q-ultraspherical polynomials. As special cases, we obtain simple closed formulas for staircas...

Abstract We settle the dual addition formula for continuous q -ultraspherical polynomials as an expansion in terms of special -Racah which constant term is given by linearization polynomials. In a second proof we derive from Rahman–Verma these using self-duality also consider limit case -Hermite

Properties of the q-ultraspherical polynomials for q being a primitive root of unity are derived using a formalism of the soq(3) algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of q-beta integrals of Ramanujan. Mathematics Subject Classifications (1991). 17B37, 33D80

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the measure |x|γ(1 − x2)1/2dx is derived which is based on a “reversing property” of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalizati...

In the present paper, the author applies some of his earlier results which extend the well-known Hille-Hardy formula for the Laguerre polynomials to certain classes of generalized hypergeometric polynomials in order to derive various generalizations of a bilinear generating function for the Jacobi polynomials proved recently by Carlitz. The corresponding results for the polynomials of Legendre,...