Orthogonal Polynomials in Mathematical Physics

Abstract

This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ($q$-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra.

46 pages

References in corpus (21)

• On AGT relation in the case of U(3)
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Matrix model version of AGT conjecture and generalized Selberg integrals
• On the complete perturbative solution of one-matrix models
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Correlators in tensor models from character calculus
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Cluster integrable systems, q-Painleve equations and their quantization
• S-duality as a beta-deformed Fourier transform
• On determinant representation and integrability of Nekrasov functions
• Eigenvalue hypothesis for multi-strand braids
• On R-matrix approaches to knot invariants
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• CFT approach to the $q$-Painlevé VI equation
• q-Painleve equation from Virasoro constraints
• Conformal blocks and Painlevé functions
• New results in $\mathcal{N}=2$ theories from non-perturbative string
• On the 6j-symbols for SL(2,C) group
• Discrete series of representations for the modular double of U_q(sl(2,R))
• Braiding Operator via Quantum Cluster Algebra

Cited by in corpus (8)

• On (q,t)-deformation of Gaussian matrix model
• Nimble evolution for pretzel Khovanov polynomials
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Lens partition function, pentagon identity and star-triangle relation
• On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$