A. U. Klimyk

Representations of the Heisenberg Group and Special Functions

The maximal nilpotent group N in U(n − 1,1) consists of matrices (6) of Section 11.1.1. Replace in these matrices a by −2c.

Relations between spherical functions of compact groups

The associated spherical functions of the homogeneous spaces SO(n)/SO(n−1), U(n)/U(n−1), and Sp(n)/Sp(n−1) are found in different coordinate systems: They are matrix elements of representation operators, right invariant with respect to the subgroups SO(n−1), U(n−1), and Sp(n−1), respectively. The left index of these matrix elements corresponds to the reductions onto the subgroups SO(p)×SO(q); U(p)×U(q); and Sp(p)×Sp(q), p+q=n. The relations between these spherical functions are derived. These relations lead to the formulas connecting the Clebsch–Gordan coefficients for the groups SO(n), U(n), and Sp(n).

h-Harmonic Polynomials, h-Hankel Transform, and Coxeter Groups

The aim of this chapter is to give Dunkl’s results on h-harmonic polynomials and the h-Hankel transform. They are related to symmetries with respect to Coxeter groups. The theory of h-harmonic polynomials is an analogue of the theory of usual harmonic polynomials. h-Harmonic polynomials are polynomials on the Euclidean space E n vanishing under action of the h-Laplacian. The analogue of majority of results from the theory of harmonic polynomials is valid for h-harmonic polynomials. The theory of h-harmonic polynomials leads to the h-Bessel function and to the h-Hankel transform.

Hypergeometric Functions Related to Jack Polynomials

Let us fix a positive number α determining Jack polynomials J λ(x (r); α). Let d = 2/α. For every partition λ = (λ 1, λ 2,...,.λ n ) we put

Clebsch-Gordan Coefficients and Racah Coefficients of Finite Dimensional Representations

\left( a \right)_\lambda ^{\left( \alpha \right)} \equiv {\left( a \right)_\lambda } = \sum\limits_{i = 1}^{l\left( \lambda \right)} {{{\left( {a - \frac{d}{2}\left( {i - 1} \right)} \right)}_{{\lambda _i}}}} ,

Gel’fand Hypergeometric Functions

(1) where (b) k = b(b+1)...(b+k−1), (b)0=1.

Special Functions Connected with SO ( n ) and with Related Groups

Irreducible representations \({T_{{m_n}}}\) of the unitary group U(n) are given (up to an equivalence) by highest weights m n = (m 1n ,... ,m nn ), where m in ∈ℤ and m 1n ≥ m 2n ≥ ... ≥ m nn . The tensor product \({T_{{m_n}}} \otimes {T_{\left( {p,0} \right)}}\) of the irreducible representations of the group U(n) with highest weights m n and (p,0) ≡ (p, 0,... , 0) decomposes into irreducible representation and every of them is contained in the decomposition not more that once.

Representations of Groups, Related to SO(n−1), in Non-Canonical Bases, Special Functions, and Integral Transforms

In Chapter 8 of the book [371] we exposed the theory of Clebsch-Gordan coefficients (CGC’s) and Racah coefficients (RC’s) of finite dimensional representations of the group SU(2). These coefficients led us to orthogonal polynomials of a discrete variable. In this chapter we represent a foundation of CGC’s and RC’s for finite dimensional representations of compact groups. These coefficients form complete orthonormal systems of functions of many discrete variables. CGC’s and RC’s of the unitary group U(n) lead to generalized multivariate hypergeometric functions. The theory of these hypergeometric functions will be given in Chapter 5.