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

In this topical review, we review the present status of the spin foam formulation of non-perturbative (background-independent) quantum gravity. The topical review is divided into two parts. In the first part, we present a general introduction to the main ideas emphasizing their motivation from various perspectives. Riemannian three-dimensional gravity is used as a simple example to illustrate conceptual issues and the main goals of the approach. The main features of the various existing models for four-dimensional gravity are also presented here. We conclude with a discussion of important questions to be addressed in four dimensions (gauge invariance, discretization independence, etc). In the second part, we concentrate on the definition of the Barrett–Crane model. We present the main results obtained in this framework from a critical perspective. Finally, we review the combinatorial formulation of spin foam models based on the dual group field theory technology. We present the Barrett–Crane model in this framework and review the finiteness results obtained for both its Riemannian and its Lorentzian variants.

http://inspirehep.net/record/612372/files/arXiv:gr-qc_0301113.pdf

Spin foam models for quantum gravity

We discuss DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory in the leading and next-to-leading approximations. Eigenvalues of the BFKL kernel in this model turn out to be analytic functions of the conformal spin. It allows us to find the residues of the anomalous dimensions of the twist-2 operators in the points j=1,0,-1, ... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. The holomorphic separability of the BFKL kernel and the integrability of the DGLAP dynamics in this model are also discussed.

DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. 
We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. 
As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.

https://iopscience.iop.org/article/10.1088/0264-9381/17/16/302/pdf

A Lorentzian signature model for quantum general relativity

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/67/3/R03/pdf

Quantum systems with finite Hilbert space

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 investigate representations of classical noncompact Lie groups induced by representations of the maximal parabolic subgroup. Formulas are obtained for 5O0(2p, 2q)/U(p, q) and U(2p, 2q) ↑ Sp(p, q).

Spectral expansions of some representations of Lie groups

In the 18-th and 19-th centuries there appeared a great number of types of special functions to solve the equations of mathematical physics and to calculate the integrals. Many of them turned out to be special or limiting cases of the hypergeometric function F(α,β;γ;x), introduced in 1769 by L. Euler and scrutinized at the beginning of the 19-th century by Gauss. Gauss’ work triggered a flow of investigations which established different recurrent relations, differential equations, integral representations, generating functions, addition and multiplication theorems, asymptotic expansions for the hypergeometric function and its associates (Legendre, Gegenbauer, Hermite, Laguerre, Chebyshev polynomials; Bessel, Neumann, Macdonald, Whittaker functions, etc.), sought for relations between these functions, and calculated puzzling integrals involving them, etc. Books containing hundreds of pages were devoted to studies of some classes of special functions.

N. Ya. Vilenkin

Papers

Representation of Lie groups and special functions

Spectral expansions of some representations of Lie groups

Symmetries and Special Functions