We sample certain results from the theory of q-series, including summation and transformation formulas, as well as some recent results which are not available in book form. Our approach is systematic and uses the Askey–Wilson calculus and Rodrigues-type formulas.

This chapter contains a survey of known results and open problems connected to the combinatorics of (type A) Macdonald polynomials. Macdonald polynomials are symmetric functions in a set of variables which depend on two extra parameters q,t. They include most of the commonly studied bases for the ring of symmetric functions, such as Schur functions and Hall-Littlewood polynomials, as special cases. Macdonald polynomials have geometric interpretations which make them important to algebraic geometry and mathematical physics, and are also fundamental to the study of special functions. Their combinatorial properties are rather mysterious, although a lot of progress has been made on the type A case in the past 20 years, in conjunction with the study of the representation theory of the ring of diagonal coinvariants. This survey shows how to express Macdonald polynomials, and other important objects such as the bigraded Hilbert series of the diagonal coinvariant ring, in terms of popular combinatorial structures including tableaux, Dyck paths, and parking functions.

The purpose of this chapter is to give an elementary introduction to the most fundamental aspects of elliptic hypergeometric functions. We will also give some indication of their historical origin in statistical mechanics. We will explain how elliptic hypergeometric functions can be used to construct biorthogonal rational functions, generalizing the famous Askey scheme of orthogonal polynomials. Apart from some general mathematical maturity, the only prerequisites will be elementary linear algebra and complex function theory.

The A-hypergeometric or GKZ hypergeometric system of differential equations in the present form were introduced by Gel'fand, Zelevinsky, and Kapranov about 30 years ago. Series solutions are multivariable hypergeometric series defined by a matrix A. They found that affine toric ideals and their algebraic and combinatorial properties describe solution spaces of the A-hypergeometric differential equations, which also opened new research areas in commutative algebra, combinatorics, polyhedral geometry, and algebraic statistics. This chapter describes fundamental facts about the system and its solutions, and also gives pointers to recent advances. Applications of A-hypergeometric functions are getting broader. Early applications were mainly to period maps and algebraic geometry. The interplay with commutative algebra and combinatorics has been a source of new ideas for these two fields and for the theory of hypergeometric functions. Recent new applications are to multivariate analysis in statistics.

By introducing weight functions on Euclidean space which are products of powers of linear functions vanishing on the mirrors of a finite reflection group one can construct generalizations of classical harmonic and Fourier analysis. There is a commutative algebra of differential-difference operators (the Dunkl operators) which generalize the partial derivatives and which are equipped with parameters. This chapter gives an introduction to the requisite properties of finite reflection (Coxeter) groups, which are associated to root systems. This is followed by the construction and commutativity proofs for the Dunkl operators. By their use one defines a concept of harmonic polynomials which serve as orthogonal bases for functions on the surface of the standard unit sphere with respect to the group-invariant weight function. There is an analog of the exponential function, the Dunkl kernel, which is used to define a generalized Fourier transform. The one-variable version of the transform is related to the classical Hankel transform. The two-dimensional examples of harmonic polynomials include the Gegenbauer and Jacobi polynomials. The general theory includes a natural extension of the classical orthogonal polynomials and of the Bessel functions.

This chapter provides an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, a number of summations, transformations, and explicit evaluations for such multiple series and integrals is listed. The focus is on those results that do not directly extend to the elliptic level. The featured results include multivariate versions of the terminating q-binomial theorem, the q-Pfaff-Saalschütz summation, the Jackson summation, some multilateral summations including multivariate versions of Dougall's 2H2 summation, Ramanujan's 1psi1 summation, Bailey's 6psi6 summation, multivariate Watson and Bailey transformations, dimension changing transformations, and multidimensional generalizations of the Askey-Wilson integral evaluation. A survey on the theory of basic hypergeometric series with Macdonald polynomial argument is provided as well.

The aim of this chapter is to introduce the formal theory of general orthogonal polynomials and present the two dual combinatorial approaches due to Foata for the special function aspects of the orthogonal polynomials, and to Flajolet and Viennot for the lattice paths models used for the moments and general orthogonal polynomials. After reviewing the standard interplay between orthogonal polynomials and combinatorics, influenced by their pioneering works, we will report on some recent topics developed in this cross-cutting field of these two branches of mathematics.

The KZ equations are a fundamental mathematical structure related to hypergeometric functions. Solutions of all versions of KZ equations are given by multidimensional hypergeometric integrals. The semi-classical limit of KZ equations leads to basic quantum chain models of mathematical physics and representation theory. In this chapter we describe the main examples of the KZ equations (rational, trigonometric, elliptic, differential or difference) with integral hypergeometric solutions. We also describe the semi-classical limit of KZ equations and associated Bethe ansatz method as the semi-classical limit of the hypergeometric solutions.

The su(2) 3j-coefficients (or symbols) and higher ones as 6j and 9j play a crucial role in various physical applications dealing with the quantization of angular momentum. In this chapter, the hypergeometric expressions for these coefficients and their relations to discrete orthogonal polynomials are emphasized. We give a short summary of the relevant class of representations of the Lie algebra su(2), and discuss their tensor product. In the tensor product decomposition, the important Clebsch-Gordan coefficients appear. 3j-Coefficients are proportional to these Clebsch-Gordan coefficients. We give some useful expressions (as hypergeometric series) and their relation to Hahn polynomials. Next, the tensor product of three representations is considered, and the relevant Racah coefficients (or 6j-coefficients) are defined. The explicit expression of a Racah coefficient as a hypergeometric series of type 4F3 and the connection with Racah polynomials and their orthogonality is given.9j-Coefficients are then defined in the context of the tensor product of four representations. They are related to a discrete orthogonal polynomial in two variables. Finally, we consider the tensor product of (n+1) representations and generalized recoupling coefficients or 3nj-coefficients, determined by two binary coupling schemes.

We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two in essence form an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integrals and series on root systems. The third and final part gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part also developed by Coskun and Gustafson). We survey the main properties of elliptic BC_n interpolation functions and BC_n-symmetric biorthogonal functions, which generalize Okounkov's BC_n interpolation Macdonald polynomials and the Koornwinder polynomials, respectively.

Appell introduced four kinds of hypergeometric series in two variables as extensions of the hypergeometric series F(a,b,c;x), and Lauricella generalized them to hypergeometric series in m variables, and they considered systems of partial differential equations satisfied by them. In this chapter, we give definitions of Appell’s and Lauricella’s hypergeometric series and state their fundamental properties such as domains of convergence, integral representations, systems of partial differential equations, fundamental systems of solutions, and transformation formulas. We define the rank and the singular locus of a system of partial differential equations, and list them for Appell’s and Lauricella’s systems. We describe Pfaffian systems, contiguity relations, monodromy representations and twisted period relations for the systems. We give their explicit forms for Lauricella’s E_D, which is the simplest among Lauricella’s systems. We also mention the uniformization of the complement of the singular locus of E_D by the projectivization of its fundamental system of solutions.

This chapter gives an overview of the theory of nonsymmetric and symmetric Macdonald-Koornwinder polynomials. The setup of the theory is new, allowing for a uniform treatment of all known cases, including a new rank two case. Among the basic properties of the Macdonald-Koornwinder polynomials discussed in the chapter are the (bi)orthogonality relations, the quadratic norm formulas, duality, and evaluation formulas. The chapter also gives an introduction to the associated theory of double affine Hecke algebras.