We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event p...

We define a family of polynomials of the form ∑ f(σ)x1,σ(1) · · ·xn,σ(n) in terms of the Kazhdan-Lusztig basis {C′ w(1) |w ∈ Sn} for the symmetric group algebra C[Sn]. Using this family, we obtain nonnegativity properties of polynomials of the form ∑ cI,I′∆I,I′(x)∆I,I′(x). In particular, we show that the application of certain of these polynomials to Jacobi-Trudi matrices yields symmetric funct...

In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual s...

In this note we give sharp Schur type inequalities for multivariate polynomials with generalized Jacobi weights on arbitrary convex domains. In particular, these results yield estimates for norms of factors of multivariate polynomials.

Let dμα,β(x) = (1−x)(1+x)dx, α, β > −1, be the Jacobi measure supported on the interval [−1, 1]. Let us introduce the Sobolev inner product

In the scalar case, it is well known that the zonal spherical functions of any compact Riemannian symmetric space of rank one can be expressed in terms of the Jacobi polynomials. The main purpose of this paper is to revisit the matrix valued spherical functions associated to the complex projective plane to exhibit the interplay among these functions, the matrix hypergeometric functions and the ...

Fourier orthogonal series with respect to the weight function (1 − |x|2)μ−1/2 on the unit ball in Rd are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourie...

We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and PoJlaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlit...

Inner products of the type 〈f, g〉S = 〈f, g〉ψ0 + 〈f ′, g〉ψ1, where one of the measures ψ0 or ψ1 is the measure associated with the Jacobi polynomials, are usually referred to as Jacobi-Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Jacobi-Sobolev inner products. The inner products are such that the associated pair...