Big q-Jacobi Polynomials Overview

• Big q-Jacobi polynomials are defined by a terminating basic hypergeometric series and exhibit discrete orthogonality on interlaced q-lattices.
• They satisfy a three-term recurrence relation and are eigenfunctions of a second-order q-difference operator, underpinning their spectral properties.
• They play a vital role in q-analysis, representation theory, and combinatorics, linking q-Racah polynomials to classical orthogonal systems.

The big $q$-Jacobi polynomials form a principal branch of the $q$-hypergeometric orthogonal polynomial hierarchy, sitting directly below the $q$-Racah family in the Askey scheme. Parameterized by $a$, $b$, $c$, and $q$ (with $0$q$-lattices. Their analytic, algebraic, and spectral properties underlie a variety of applications in $q$-analysis, representation theory of quantum algebras, and analytic combinatorics, and they serve as a canonical example connecting discrete and continuous orthogonal polynomial systems.

1. Definition and Hypergeometric Series Representation

The big $q$-Jacobi polynomials $P_{n}(x;a,b,c;q)$ are defined by a terminating basic hypergeometric series: $$P_{n}(x;a,b,c;q) = {}_3\phi_2 \left( \begin{matrix} q^{-n}, ab\,q^{n+1}, x \ a\,q,\;c\,q \end{matrix} ; q,\;q \right) = \sum_{k=0}^n \frac{(q^{-n};q)_k (ab\,q^{n+1};q)_k (x;q)_k}{(a\,q;q)_k (c\,q;q)_k (q;q)_k} q^k,$$ where $(\alpha;q)_k = \prod_{j=0}^{k-1} (1-\alpha q^j)$ is the $q$-Pochhammer symbol. The series truncates at $k=n$ due to the numerator parameter $q^{-n}$, ensuring polynomiality in $x$ of degree $n$ (Odake et al., 2016, Baseilhac et al., 2018, Koornwinder, 2010, Costas-Santos et al., 2010, Bos et al., 20 Jan 2026, Alvarez-Nodarse et al., 2011, Liu, 2018, Chern et al., 2023).

Parameter admissibility for orthogonality and regularity commonly requires $a q,\,b q<1$, $c<0$ and $0

2. Orthogonality, Weight Functions, and Discrete Measures

Big $q$-Jacobi polynomials admit a discrete orthogonality with support on two $q$-geometric progressions. The prototypical orthogonality relation is given via a Jackson-type $q$-integral or an equivalent discrete sum: $$\int_{cq}^{aq} P_m(\eta) P_n(\eta)\, w(\eta)\; d_q \eta = h_n\, \delta_{mn},$$ where

$$w(\eta) = \frac{(\eta,\,b c^{-1} \eta;q)_\infty}{(a^{-1}\eta,\,c^{-1}\eta;q)_\infty},$$

and the $q$-integral

$$\int_{cq}^{aq} f(\eta)\, d_q \eta = (1-q)\!\!\sum_{k=0}^\infty \!\!\left[a\,q^{k+1} f(a q^{k+1}) - c\,q^{k+1} f(c q^{k+1})\right].$$

Alternatively, the orthogonality decomposes as a two-component sum over the two $q$-lattices $\{ a\,q^{k+1} \}_{k=0}^\infty$ and $\{ c\,q^{k+1} \}_{k=0}^\infty$ with suitable weights.

The squared norm $h_n$ (for example, as given in (Odake et al., 2016)) is an explicit product of $q$-shifted factorials: $$h_n = (a c q^2)^{-n} q^{-n(n-1)} \frac{(a q, b q, c q, ab c^{-1} q;q)_n}{(q, ab q^2, a c^{-1} q, b c q;q)_n} \frac{(a q, b q, c q, ab c^{-1} q;q)_\infty}{(q, ab q^2, a c^{-1} q, b c q;q)_\infty}.$$ Orthogonality holds for all $m, n \geq 0$ in the regime $0<aq,\,bq<1$, $c<0$, $0Odake et al., 2016, Costas-Santos et al., 2010, Koornwinder, 2010, Liu, 2018).

At roots of unity ($q^N=1$), a finite system of polynomials is orthogonal on the roots of $P_{N+1}(x)$ with explicit mass point weights (Costas-Santos et al., 2010).

3. Recurrence Relations and $q$-Difference Operators

The big $q$-Jacobi polynomials satisfy a three-term recurrence relation: $$x P_n(x) = A_n P_{n+1}(x) + B_n P_n(x) + C_n P_{n-1}(x),\qquad n\ge 1,$$ with explicit coefficients. For instance, in the standard normalization (Odake et al., 2016): $$A_n = -\frac{(1-aq^{n+1})(1-abq^{n+1})(1-cq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})},$$

$$C_n = a c q^{n+1} \frac{(1-q^{n+1})(1-bq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})},$$

and $B_n = -A_n - C_n$, with $P_{-1}\equiv0$.

There exist alternative, yet equivalent, forms for these coefficients varying with the normalization and literature conventions (Koornwinder, 2010, Baseilhac et al., 2018, Alvarez-Nodarse et al., 2011). All such forms encode the spectrum and combinatorial structure of the big $q$-Jacobi family.

In operator terms, $P_n(x)$ are eigenfunctions of a second order $q$-difference (difference Schrödinger) operator $H$ with explicit $x$-dependent coefficients: $$H P_n(x) = E_n P_n(x),\qquad E_n = q^{-n-1}(1-ab q^{n+1}) \quad[1604.00714], [1806.02656].$$ In the representation-theoretic realization, this operator corresponds to an element of the Askey-Wilson algebra, and $P_n(x)$ diagonalize certain tridiagonalized images of $U_q(\mathfrak{sl}_2)$ generators (Baseilhac et al., 2018).

4. Connections to Representation Theory, Algebraic Structures, and Limit Relations

Big $q$-Jacobi polynomials naturally arise as basis elements in representations of the Askey-Wilson algebra and in the study of twisted primitive elements of quantum groups $U_q(\mathfrak{sl}_2)$ (Baseilhac et al., 2018). The polynomials can be identified as the basis vectors diagonalizing a tridiagonal operator within the Askey-Wilson algebra, with the parameter triple $(a,b,c)$ reflecting module labels or structure constants.

A key algebraic feature is the closure relation satisfied by the multiplication operator in the polynomial basis,

$$[H,[H,\eta]] = \eta R_0(H) + [H,\eta] R_1(H) + R_{-1}(H),$$

where $R_0, R_1, R_{-1}$ are polynomials in the Hamiltonian $H$. This structural property enables the derivation of Heisenberg operator solutions and the construction of explicit "creation/annihilation" operators acting on the polynomial space (Odake et al., 2016).

In the broader Askey scheme, the big $q$-Jacobi polynomials are the uniform analytic continuation of $q$-Racah polynomials under the limit $N\to\infty$, preserving both orthogonality and recurrence structure (Koornwinder, 2010, Alvarez-Nodarse et al., 2011). Setting $c\to 0$ recovers the little $q$-Jacobi polynomials, and classical Jacobi polynomials arise in the $q\to1^{-}$ limit.

Table: Principal Relationships within the $q$-Askey Scheme

Family	Limiting/Parameter Regimes	Resulting Family
$q$-Racah	$N\to\infty$	Big $q$-Jacobi
Big $q$-Jacobi	$c\to0$	Little $q$-Jacobi
Big $q$-Jacobi	$q\to 1^-$, affine rescaling	Jacobi (classical)
Big $q$-Jacobi	$ab = q^{-N}$ or $c=q^{-N}$	$q$-Hahn/Dual $q$-Hahn

5. Extended Self-Adjointness, Hilbert Space Structures, and Spectral Theory

Standard operator analysis shows unbounded Jacobi (tridiagonal) matrices representing big $q$-Jacobi recurrences are not self-adjoint on a single $\ell^2$ chain due to non-vanishing off-diagonal coefficients at infinity. The construction of a self-adjoint Hamiltonian is enabled by extending the Hilbert space to $\ell^2\oplus\ell^2$, associating the two components to the two $q$-chains $a\,q^{k+1}$ and $c\,q^{k+1}$ (Odake et al., 2016).

In this setup, each sector admits a ground state, and the orthonormal basis for the full space is constructed using both sets of polynomials. The Hamiltonian becomes self-adjoint with respect to

$$(f,g) = \sum_{x\ge0} \left[ f_+(x)g_+(x) + f_-(x)g_-(x) \right],$$

with $f_{\pm}(x)$ the two components. This formulation is essential for spectral completeness and the construction of a fully orthogonal eigenbasis (Odake et al., 2016).

6. Applications and Recent Developments

Big $q$-Jacobi polynomials appear in numerous analytic and algebraic applications:

• $q$-Euler numbers and Hankel determinants: Specializations of the big $q$-Jacobi polynomials yield the Favard orthogonal system for the $q$-Euler numbers, allowing continued fraction expansions for the $q$-Euler generating functions and closed-form evaluations of Hankel determinants (Chern et al., 2023).
• Subdivision schemes and Chebyshev reciprocals: Recent identities establish that big $q$-Jacobi polynomials, at specific parameters, are reciprocals of Chebyshev polynomials of the first kind, leading to explicit symbols for exponential-reproducing subdivision schemes (Bos et al., 20 Jan 2026).
• $q$-Beta and Nassrallah-Rahman integrals: The integral representations associated to their orthogonality produce $q$-beta integrals subsuming classical (Askey-Wilson, Nassrallah-Rahman) $q$-integrals, and under pinning "strange" $q$-series summations (Liu, 2018).
• Limit relations and factorization: Degenerate cases correspond to $q$-Hahn, dual $q$-Hahn, big $q$-Laguerre, and Al-Salam–Carlitz I polynomials, with explicit factorization and orthogonality preserved under limiting procedures (Alvarez-Nodarse et al., 2011, Costas-Santos et al., 2010, Koornwinder, 2010).

7. Summary of Main Properties

The following table encapsulates the central features of the big $q$-Jacobi polynomials.

Feature	Formula / Description	Source
Definition	$${}_3\phi_2 \left( q^{-n}, abq^{n+1}, x; aq, cq; q, q\right)$$	(Odake et al., 2016, Koornwinder, 2010)
Orthogonality	Discrete, two $q$-lattice, Jackson-type sum/integral	(Odake et al., 2016, Costas-Santos et al., 2010)
Recurrence	$x P_n = A_n P_{n+1} + B_n P_n + C_n P_{n-1}$	(Odake et al., 2016, Baseilhac et al., 2018)
$q$-Difference Operator	Second-order; explicit action, closure relation	(Odake et al., 2016, Baseilhac et al., 2018)
Limit transitions	$q$-Racah $\to$ big $q$-Jacobi $\to$ little $q$-Jacobi, etc.	(Koornwinder, 2010, Alvarez-Nodarse et al., 2011)
Representation Theory	Askey-Wilson algebra embedding, $U_q(\mathfrak{sl}_2)$ modules	(Baseilhac et al., 2018)
Analytic Applications	Subdivision schemes, generating functions, $q$-Euler/Hankel	(Bos et al., 20 Jan 2026, Chern et al., 2023)

The big $q$-Jacobi polynomials thus serve as a keystone in the hierarchy of $q$-orthogonal polynomials, with explicit realizations connecting operator theory, $q$-special functions, and combinatorics, and with precise structural and spectral properties underpinning their diverse applications (Odake et al., 2016, Baseilhac et al., 2018, Bos et al., 20 Jan 2026, Alvarez-Nodarse et al., 2011, Chern et al., 2023, Koornwinder, 2010, Costas-Santos et al., 2010, Liu, 2018).