Definition
q‑Racah polynomials are a family of basic (or q‑) hypergeometric orthogonal polynomials that occupy the top level of the Askey–Wilson scheme of hypergeometric orthogonal polynomials. They can be expressed in terms of the terminating balanced basic hypergeometric series ${}_4\phi_3$ and satisfy a discrete orthogonality relation on a finite set of points.
Overview
Introduced in the 1980s as a q‑analogue of the classical Racah polynomials, q‑Racah polynomials serve as a unifying object for many families of orthogonal polynomials. They appear in diverse areas such as representation theory of quantum groups, exactly solvable models in statistical mechanics, and the theory of basic hypergeometric functions. Their parameters allow specialization to all lower‑level families in the Askey–Wilson hierarchy, including the Askey–Wilson, continuous q‑Jacobi, and q‑Hahn polynomials.
Etymology / Origin
The name combines “Racah”, after the Racah coefficients (or 6‑j symbols) in the theory of angular momentum, with the prefix “q‑” indicating a deformation parameter $q$ (typically $0<|q|<1$) that replaces ordinary integers by their q‑analogues. The polynomials were first systematically studied by R. Askey and J. Wilson and later by G. Baxter, K. Koekoek, R. Swarttouw, and others.
Characteristics
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Explicit Form
$$ R_n\bigl(\mu(x);a,b,c,d\mid q\bigr)= {}_4\phi_3!\left(\begin{matrix} q^{-n},;abcd,q^{n-1},;q^{-x},;cd,q^{x+1}$$2pt] ab,q,;ac,q,;ad,q \end{matrix},\bigg|,q,q\right), $$ where $\mu(x)=q^{-x}+cd,q^{x+1}$ and $a,b,c,d$ are complex parameters subject to certain non‑degeneracy conditions. -
Orthogonality
They satisfy a finite discrete orthogonality relation
$$ \sum_{x=0}^{N} w(x),R_m(\mu(x)),R_n(\mu(x)) = h_n,\delta_{mn}, $$ where the weight $w(x)$ and norm $h_n$ are explicit rational functions of the parameters and $q$. -
Recurrence and Difference Equations
q‑Racah polynomials obey a three‑term recurrence relation in the degree $n$ and a second‑order q‑difference equation in the variable $x$. Both relations reflect the bispectral property typical of orthogonal polynomial families. -
Limits and Specialisations
By appropriate limiting processes (e.g., letting one parameter tend to zero or $q\to1$), q‑Racah polynomials reduce to:- Racah polynomials ($q\to1$),
- q‑Hahn, q‑Krawtchouk, and dual q‑Krawtchouk polynomials,
- Askey–Wilson polynomials (by continuous rather than discrete argument).
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Applications
- Quantum algebra: matrix elements of $U_q(\mathfrak{su}(2))$ representations and q‑Racah coefficients.
- Statistical mechanics: solutions of the Yang–Baxter equation and exactly solvable lattice models.
- Combinatorics: enumeration of certain weighted paths and connections with q‑binomial coefficients.
Related Topics
- Askey–Wilson polynomials – the most general continuous family in the hierarchy.
- Racah coefficients (6‑j symbols) – classical angular‑momentum coupling coefficients whose q‑analogue is linked to q‑Racah polynomials.
- Basic hypergeometric series – the series ${}_r\phi_s$ providing the analytic foundation of q‑polynomials.
- Quantum groups – algebraic structures where q‑Racah polynomials appear as representation‑theoretic objects.
- The Askey scheme – a classification diagram of hypergeometric orthogonal polynomials, with the q‑Racah family at its apex.