Representation of Lie Groups and Special Functions: Recent Advances

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This text is a continuation of the three-volume work Representation of Lie Groups and Special Functions by the same authors. Here, they deal with the exposition of the main new developments in the contemporary theory of multivariate special functions. The theory of orthogonal symmetric polynomials (Jack polynomials, Macdonald’s polynomials and others) and multivariate hypergeometric functions associated to symmetric polynomials are treated. Multivariate hypergeometric functions, multivariate Jacobi polynomials and h-harmonic polynomials connected with root systems and Coxeter groups are introduced. Also, the theory of Gel'fand hypergeometric functions and the theory of multivariate hypergeometric series associated to Clebsch-Gordan coefficients of the unitary group U(n) is given. The volume concludes with an extensive bibliography. This text is intended for research mathematicians and physicists, postgraduate students in mathematics and mathematical and theoretical physics.