Group representations in probability and statistics

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

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic functions.

Representation of Lie groups and special functions

We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs. The bounds compare well with exact answers for a variety of simple chains and seem better than bounds derived through Cheeger-like inequalities. They offer improved rates of convergence for the random walk associated to approximate computation of the permanent.

Geometric Bounds for Eigenvalues of Markov Chains

The cyclic sieving phenomenon

New statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to thet-core of a partition. Usingq-series, some explicit formulas are given for the number of partitions which aret-cores. Some related questions for self-conjugate and distinct partitions are discussed.

Cranks and t-cores.

The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilson's 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehler's formula.

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

A schensted algorithm for rim hook tableaux

Many evaluations of terminating hypergeometric series at arguments other than 1 are given. Some are equivalent to some unpublished work of Gosper, while others are new. In particular, two new evaluations of ${}_7 F_6 $’s with four parameters are stated. The main technique is a change of variables formula which is equivalent to the Lagrange inversion formula. A new proof of Whipple’s transformation of a very well poised ${}_7 F_6 $ into a Saalschutzian ${}_4 F_3 $ is a corollary.

Dennis Stanton

Papers

The cyclic sieving phenomenon

Cranks and t-cores.

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

A schensted algorithm for rim hook tableaux

Strange Evaluations of Hypergeometric Series