Abstract. The irreducible * -representations of the Lie algebra su(1, 1) consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for Uq(su(1, 1)), which turn out to be Askey-Wilson functions and Askey-Wilson polynomials. IntroductionIn this paper we study Racah coefficients, or 6j-symbols, for representations of the Lie algebra su(1, 1). The Racah coefficients for certain tensor product representations of su(1, 1) lead to unitary integral transforms with a very-well-poised 7 F 6 -function, called a Wilson function, as a kernel. These Wilson functions and the corresponding integral transforms were recently introduced by the author in [12] with the applications in this paper in mind. We also consider a q-analogue of su(1, 1), namely the quantized universal enveloping algebra U q (su (1, 1)). From the Racah coefficients for certain tensor product representations of U q (su (1, 1)) we obtain a new interpretation of the Askey-Wilson functions transform introduced by Koelink and Stokman [23].As an application we obtain several identities for special functions involving (Askey-) Wilson functions and polynomials.Racah coefficients for su(2) play an important role in the theory of angular momentum in quantum physics [7]. They were first studied in the 1940's by Racah [32], who also obtained an explicit expression as a finite single sum for these coefficients. Only much later it was realized that the Racah coefficients are multiples of polynomials of hypergeometric type, so that the orthogonality relations for the Racah coefficients lead to discrete orthogonality relations for certain polynomials. These polynomials are nowadays called the Racah polynomials, which can be defined bywhere one of the lower parameters is equal to −N , N ∈ Z ≥0 , and 0 ≤ n ≤ N . These are polynomials of degree n in the variable x(x + γ + δ + 1), and they are orthogonal on the set {0, 1, . . . , N };

The quantum group analogue of the normalizer of SU (1, 1) in SL(2, C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U q (su(1, 1)). It turns out that U q (su(1, 1)) does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analogue of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analogue of the normalizer of SU (1, 1) in SL(2, C) is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to U q (su(1, 1))-representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately.The paper is split into two parts; the first part gives almost all of the statements and the results, and the statements in the first part are independent of the second part. The second part contains the proofs of all the statements.

Abstract. The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra su(1, 1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and the discrete terms are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch-Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.