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Wilson function transforms related to Racah coefficients

Wolter Groenevelt

TL;DR

<3-5 sentence high-level summary> The paper establishes a deep link between Racah coefficients for representations of $\mathfrak{su}(1,1)$ and Wilson functions, showing that 6j-symbols arise as kernels of unitary Wilson transforms of type I and II. It provides explicit expressions for Clebsch–Gordan and Racah coefficients in terms of Wilson polynomials and continuous dual Hahn polynomials, and extends the framework to the quantum group $\mathcal{U}_q(\mathfrak{su}(1,1))$, where Askey–Wilson functions and polynomials play the analogous role. The constructions yield generalized summation and integral identities that unify recoupling coefficients with hypergeometric orthogonal polynomials, and reveal BE-type identities as manifestations of these transforms. These results offer new interpretations and tools for spectral analysis in representation theory and hypergeometric function theory, with potential applications to $q$-deformations and special-function identities.

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 $U_q(\su(1,1))$, which turn out to be Askey-Wilson functions and Askey-Wilson polynomials.

Wilson function transforms related to Racah coefficients

TL;DR

<3-5 sentence high-level summary> The paper establishes a deep link between Racah coefficients for representations of and Wilson functions, showing that 6j-symbols arise as kernels of unitary Wilson transforms of type I and II. It provides explicit expressions for Clebsch–Gordan and Racah coefficients in terms of Wilson polynomials and continuous dual Hahn polynomials, and extends the framework to the quantum group , where Askey–Wilson functions and polynomials play the analogous role. The constructions yield generalized summation and integral identities that unify recoupling coefficients with hypergeometric orthogonal polynomials, and reveal BE-type identities as manifestations of these transforms. These results offer new interpretations and tools for spectral analysis in representation theory and hypergeometric function theory, with potential applications to -deformations and special-function identities.

Abstract

The irreducible -representations of the Lie algebra 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 , which turn out to be Askey-Wilson functions and Askey-Wilson polynomials.

Paper Structure

This paper contains 24 sections, 30 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Representations of $\mathfrak{su}(1,1)$ and Racah polynomials
  3. The Lie algebra $\mathfrak{su}(1,1)$
  4. Clebsch-Gordan coefficients for ${\pi^+ \otimes \pi^+}$
  5. Racah coefficients for $\pi^+ \otimes \pi^+ \otimes \pi^+$
  6. Clebsch-Gordan coefficients
  7. Clebsch-Gordan coefficients for $\pi^+ \otimes \pi^-$
  8. Clebsch-Gordan coefficients for $\pi^+ \otimes \pi^P$
  9. Clebsch-Gordan coefficients for $\pi^P \otimes \pi^-$
  10. Racah coefficients for $\pi^+ \otimes \pi^- \otimes \pi^+$
  11. Racah coefficients
  12. The Wilson function transform of type I
  13. Racah coefficients for ${\pi^+ \otimes \pi^P \otimes \pi^-}$
  14. Racah coefficients
  15. The Wilson function transform of type II
  16. ...and 9 more sections

Key Result

Proposition 2.1

For $p\in \mathbb Z_{\geq 0}$, the operator $I: \mathcal{H}^p \rightarrow \mathbb C^{p+1}(W(x;k_1,k_2))$ defined by is unitary and intertwines $\pi^+_{k_1} \otimes \pi^+_{k_2}(\Delta(\Omega))$ with $M_{(k_1+k_2+x)(1-k_1-k_2-x)}$.

Theorems & Definitions (54)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 4.1
  • ...and 44 more