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 ...

Abstract. It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. In this paper we extend this result and show that every system of polynomials satisfying some (2N + 1)-term recurrence re...

The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same ...

Let be a positive measure on the real line, with orthogonal polynomials fpng and leading coe¢ cients f ng. The Geronimus type identity 1 jIm zj Z 1 1 P (t) jzpn (t) pn 1 (t)j dt = n 1 n Z P (t) d (t) ; valid for all polynomials P of degree 2n 2 has known analogues within the theory of orthogonal rational functions, though apparently unknown in the theory of orthogonal polynomials. We present ne...

Orthogonal polynomials on the standard simplex Σ in R are shown to be related to the spherical orthogonal polynomials on the unit sphere S in R that are invariant under the group Z2×· · ·×Z2. For a large class of measures on S cubature formulae invariant under Z2 × · · · × Z2 are shown to be characterized by cubature formulae on Σ. Moreover, it is also shown that there is a correspondence betwe...