For any sequence of polynomials {pk(t)} in one real or complex variable, where pk has degree k for k≥0, we find explicit expressions and recurrence relations infinite matrices whose entries are the numbers d(n,m,k), called linearization coefficients, that satisfypn(t)pm(t)=∑k=0n+md(n,m,k)pk(t),n,m≥0. pair polynomial sequences {uk(t)} e(n,m,k) satisfypn(t)pm(t)=∑k=0n+me(n,m,k)uk(t),n,m≥0. We als...

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 explici...

We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials depending only on the abstract combinatorial structure of the graph. Then we compare the shadow-state sums and the state-sums based on R-matrices and Clebsch...