R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

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

http://www.mittag-leffler.se/sites/default/files/IML-0405s-18.pdf

Advanced Determinant Calculus: A Complement

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their “Type II” integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E_7.

http://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p03-p.pdf

Transformations of elliptic hypergeometric integrals

Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

Elliptic hypergeometric series on root systems

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson’s biorthogonal 10
 W
 9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.

An elementary approach to 6j -symbols (classical, quantum, rational, trigonometric, and elliptic)

An Izergin--Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices

We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems

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Elliptic determinant evaluations and the Macdonald identities for affine root systems

We study polynomials of several variables which occur as coupling coefficients for the analytic continuation of the holomorphic discrete series of SU(1,1). There are three types of such polynomials, one corresponding to each conjugacy class of one-parameter subgroups. They may be viewed as multivariable generalizations of Hahn, Jacobi, and continuous Hahn polynomials and include many orthogonal and biorthogonal families occurring in the literature. We give a simple and unified approach to these polynomials using the group theoretic interpretation. We prove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for the Heisenberg group, leading to multivariable generalizations of Krawtchouk and Hermite polynomials.

Hjalmar Rosengren

Papers

Elliptic hypergeometric series on root systems

An elementary approach to 6j -symbols (classical, quantum, rational, trigonometric, and elliptic)

An Izergin--Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices

Elliptic determinant evaluations and the Macdonald identities for affine root systems

Multivariable orthogonal polynomials and coupling coefficients for discrete series representations