Closed-form evaluation of the finite sum in mixed su_q(1,1)–su_q(2) Clebsch–Gordan coefficients for κ′ = j − κ > 0

Determine a closed-form expression for the finite sum that arises in the formula for the Clebsch–Gordan coefficient ⟨κ μ, j m | κ′ μ′⟩_q of the mixed tensor product D^{κ+} ⊗ D^{j} when κ′ = j − κ > 0. Specifically, evaluate S(j, κ, m, μ, μ′; q) = ∑_{r = max{0, j − 2κ − m}}^{2j − 2κ} ([2κ + r]_q! [μ′ − j + κ − 1 + r]_q!) / ([r]_q! [m − j + 2κ + r]_q!) · q^{−(μ + κ + 1) r}, where q ∈ (0, 1), j ∈ {0, 1/2, 1, …}, m ∈ {−j, −j + 1, …, j}, κ ∈ {0, 1/2, 1, …}, and μ ∈ {κ + 1, κ + 2, …}. Equivalently, after the substitution r = j − 2κ − m + s (when j − 2κ − m > 0), determine a closed form for the transformed sum. The goal is to express the Clebsch–Gordan coefficient in this case without an unevaluated finite sum, for example in terms of symmetric q-factorials or terminating symmetric q-hypergeometric series.

Background

Section 5 studies Clebsch–Gordan coefficients for mixed representations of su_q(1,1), specifically the tensor product D^{κ+} (positive discrete series, infinite-dimensional) with D^{j} (finite-dimensional su_q(2) representation). General expressions are derived and many special values are obtained in closed form using terminating symmetric q-hypergeometric series and summation identities.

When κ′ = κ − j > 0, the coefficient simplifies to a closed form. However, in the complementary case κ′ = j − κ > 0, the coefficient reduces to a finite sum whose closed evaluation was not obtained. The authors outline two equivalent summation forms (one after a variable change) and state explicitly that a closed form was not found. Resolving this would complete the catalogue of special-value closed forms for mixed tensor products and potentially reveal additional structure linking these coefficients to known q-special functions.