3nj-symbols and identities for q-Bessel functions

The 6j-symbols for representations of the q-deformed algebra of polynomials on SU(2) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols

Fourier transforms related to a root system of rank 1.

Abstract : We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type $(C^\vee_1, C_1)$ . We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra $\mathcal H$ , and the corresponding Fourier transforms. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions

Orthogonal Stochastic Duality Functions from Lie Algebra Representations

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between ∗-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and su(1,1). Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.Analysi

Tensor product representations and special functions

Electrical Engineering, Mathematics and Computer Scienc

3nj-symbols and identities for q-Bessel functions

The 6j-symbols for representations of the q-deformed algebra of polynomials on SU(2) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols.Analysi

Orthogonal functions related to Lax pairs in Lie algebras

We study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for sl(d+ 1 , C) is studied in certain representations.Analysi

An Askey–Wilson Algebra of Rank 2

An algebra is introduced which can be considered as a rank 2 extension of the Askey–Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra Uq (sl(2, C)). It is shown that bivariate q-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding q-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate q-Racah polynomials.Analysi

Q−orthogonal dualities for asymmetric particle systems

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP(q, θ), asymmetric exclusion process, with a repulsive interaction, allowing up to θ ∈ N particles in each site, and the ASIP(q, θ), θ ∈ R+, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate q−analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP(q, θ).Analysi