Using divided differences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided differences on some distinguished symmetric polynomials (P̃ polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials...

Orthogonal polynomials, unless they are classical, require special techniques for their computation. One of the central problems is to generate the coefficients in the basic three-term recurrence relation they are known to satisfy. There are two general approaches for doing this: methods based on moment information, and discretization methods. In the former, one develops algorithms that take as...

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of sp...

Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.

We discuss formal orthogonal polynomials with respect to a moment matrix that has no structure whatsoever. In the classical case the moment matrix is often a Hankel or a Toeplitz matrix. We link this to block factorization of the moment matrix and its inverse, the block Hessenberg matrix of the recurrence relation, the computation of successive Schur complements and general subspace iterative m...

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients....

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into ...

When computingrecursively the members of a family of formal orthogonal polynomials, a division by zero can occur, thus producing a breakdown in the algorithm which has to be stopped. In this paper, such breakdowns are analyzed in detail and classiied. It is also showed how to avoid them in some particular cases. Applications to Pad e approximation, extrapolation methods and Lanczos method for s...