Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With ${\mathsf r}(d,n):= \binom{n+d-1}{n}$, the first method generates, for each $m \ge n$, two families of tight frames in ${\mathbb R}^{{\mathsf r}(d,n)}$ with ${\mathsf r}(d+1,m)$ elements. The second method generates a tight frame in ${\mathbb R}^{{\mathsf r}(d,N)}$ with $1 + N \times{\mathsf r}(d+1, N)$ vectors. All frame elements are given in explicit formulas

Convolutions for orthogonal polynomials from Lie and quantum algebra representations

The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations of the convolution identities for these polynomials. Using the Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi polynomials are obtained. From the quantised universal enveloping algebra for su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah coefficients. For the quantised universal enveloping algebra for su(2) q-Racah polynomials are interpreted as Clebsch-Gordan coefficients, and the linearisation coefficients for a two-parameter family of Askey-Wilson polynomials are derived.Comment: AMS-TeX, 31 page

Addition formulas for q-special functions

A general addition formula for a two-parameter family of Askey-Wilson polynomials is derived from the quantum $SU(2)$ group theoretic interpretation. This formula contains most of the previously known addition formulas for $q$-Legendre polynomials as special or limiting cases. A survey of the literature on addition formulas for $q$-special functions using quantum groups and quantum algebras is given

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials

Combinatorial proofs of some limit formulas involving orthogonal polynomials

AbstractThe object of this paper is to prove combinatorially several (13 of them) limit formulas relating different families of hypergeometric orthogonal polynomials in Askey's chart classifying them. We first find a combinatorial model for Hahn polynomials which, as pointed out by Foata at the ICM (1983), “contains” models for Jacobi, Meixner, Krawtchouk, Laguerre and Charlier polynomials. Seven limit formulas are proved by “looking at surviving structures” when taking the limit. A simple model, T-structures, is then used to prove (using a different technique) four more limit formulas involving Meixner-Pollaczek, Krawtchouk, Laguerre, Charlier and Hermite polynomials. The theory of combinatorial octopuses (of F. Bergeron) is recalled and two more limits are demonstrated using new models of Meixner-Pollaczek, Laguerre, Gegenbauer and Hermite polynomials