I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials

Symmetric functions and Hall polynomials

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs

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The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

The quantum group SU(2)q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed

https://iopscience.iop.org/article/10.1088/0305-4470/22/21/020/pdf

On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

/pdf/a-transplantation-theorem-for-jacobi-coefficients-34izmeuec3.pdf

A transplantation theorem for Jacobi coefficients.

Hausdorff's moment problem and expansions in Legendre polynomials☆

In much of Schoenberg's work there has been a strong interconnection between analytic and geometric reasoning. Here we use a remark he made about imbeddings of metric spaces to prove part of a conjecture about when a Jacobi polynomial Pn(x) can be expanded in terms of another P^\\x) with nonnegative coefficients. Also we get from a different special case of this conjecture some nonimbedding theorems for projective spaces. P£*(#), the Jacobi polynomial of degree n, order (a, /3), a, /3> — 1, is defined by

/pdf/jacobi-polynomial-expansions-with-positive-coefficients-and-3qzob0g3im.pdf

Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces

A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.

The q-harmonic oscillator and an analog of the Charlier polynomials

The inequalities of Vietoris have been a good source of problems and new results. Some of these are outlined, and a hypergeometric sum suggested by one of the problems is evaluted.

Richard Askey

Papers

A transplantation theorem for Jacobi coefficients.

Hausdorff's moment problem and expansions in Legendre polynomials☆

Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces

The q-harmonic oscillator and an analog of the Charlier polynomials

Vietoris’s Inequalities and Hypergeometric Series