Generalized Wilson Polynomials: Theory & Applications

• Generalized Wilson polynomials are hypergeometric orthogonal polynomials defined via a terminating 4F3 series and extended by parameter shifts and q-deformations.
• They exhibit bispectrality through three-term recurrences and difference equations, underpinning connections with degenerate Sklyanin algebras in integrable systems.
• Applications span generating functions, multivariate extensions, random matrix theory, and superintegrable models, unifying classical and modern frameworks.

Generalized Wilson polynomials comprise a rich class of hypergeometric orthogonal polynomials encompassing the classical Wilson polynomials, their multivariate analogues, $q$-deformations (notably, Askey--Wilson polynomials), and a variety of parameter-shifted and connection families. These objects serve as ultimate members of the Askey scheme, central in the analysis of bispectrality, integrable systems, and random matrix theory. Recent advances have elucidated their generating functions, connection formulas, underlying algebraic structures (notably degenerate Sklyanin algebras), and applications to superintegrable models.

1. Definition and Hypergeometric Representations

The Wilson polynomial $W_n(x^2;a,b,c,d)$ is defined for parameters $a, b, c, d$ (typically with positive real parts or in complex conjugate pairs) via the symmetric terminating ${}_4F_3(1)$ hypergeometric series: $$W_n(x^2; a,b,c,d) = (a+b)_n (a+c)_n (a+d)_n\; {}_4F_3\left( \begin{matrix} -n,\,n+a+b+c+d-1,\,a+ix,\,a-ix\ a+b,\,a+c,\,a+d \end{matrix}\biggr| 1\right)$$ where $(\alpha)_n$ is the Pochhammer symbol. The spectrum parameter is $x\in\mathbb{R}$ or $x\in(0,\infty)$.

Generalizations include:

• Parameter-shifted and connection families: Wilson polynomials with nontrivial parameter dependencies or shifted in one parameter, resulting in generating function expansions with hypergeometric ${}_7F_6$ coefficients (Cohl et al., 2018, Baeder et al., 2014).
• Askey--Wilson polynomials: The $q$-deformation encompasses an extra parameter $q$ and is defined as

$$p_n(x;a,b,c,d \mid q) = a^{-n}(ab,ac,ad;q)_n\;{}_4\phi_3\left( \begin{matrix} q^{-n}, q^{n-1}abcd, aw, aw^{-1} \ ab, ac, ad \end{matrix}; q, q \right)$$

where $x = \frac{1}{2}(w + w^{-1})$ and $(t;q)_n$ denotes the $q$-Pochhammer symbol (Liu, 2018, Cohl et al., 2024).

• Multivariable Wilson polynomials: Symmetric functions on a set of spectral variables $(x_1, ... , x_N)$ indexed by a partition $R$, given by

$$\mathbf{W}_R(\boldsymbol{x}; a,b,c,d) = \frac{1}{\Delta(\boldsymbol{x})} \det_{1\leq i,j\leq N}\left[ \frac{W_{R_j+N-j}(x_i; a,b,c,d)}{(a+b+c+d+R_j+N-j-1)_{R_j+N-j}} \right]$$

where $\Delta(\boldsymbol{x})$ is the Vandermonde determinant (Mishnyakov, 2024).

2. Orthogonality, Weight Measures, and Recurrence

Single-variable orthogonality: $$\int_0^\infty W_n(x^2;a,b,c,d) W_m(x^2;a,b,c,d)\, w(x)\,dx = h_n(a,b,c,d)\,\delta_{nm}$$ with

$$w(x) = \left| \frac{ \Gamma(a+ix)\Gamma(b+ix)\Gamma(c+ix)\Gamma(d+ix) }{ \Gamma(2ix) } \right|^2$$

and $h_n(a,b,c,d)$ given explicitly in terms of products of Gamma functions (Gisonni et al., 2020, Mishnyakov, 2024).

Multivariable orthogonality: For vectors $\boldsymbol{x}$,

$$\int_{(0,\infty)^N} \left[ \prod_{i=1}^N w(x_i)\right]\, \Delta(\boldsymbol x)^2\, \mathbf{W}_R(\boldsymbol x) \mathbf{W}_Q(\boldsymbol x) \, d^N x = \delta_{RQ} H^{(N)}_R(a,b,c,d)$$

with $H^{(N)}_R$ factorizing over single-variable norms (Mishnyakov, 2024).

Three-term recurrence (single-variable): $$x^2 W_n(x) = W_{n+1}(x) + B_n W_n(x) + C_n W_{n-1}(x)$$ with $B_n, C_n$ rational functions of $n$ and the parameters, matching the general bispectral structure (Bergeron et al., 2021, Mishnyakov, 2024).

Difference equation: $$Q f(x) = \frac{(ix+a)(ix+b)(ix+c)(ix+d)}{2ix(2ix-1)} f(x+1) + \frac{(-ix+a)(-ix+b)(-ix+c)(-ix+d)}{-2ix(2ix+1)} f(x-1) - [\cdots] f(x)$$ and $Q W_n(x^2;a,b,c,d) = \lambda_n W_n(x^2;a,b,c,d)$ with $\lambda_n = n(n+a+b+c+d-1)$ (Bergeron et al., 2021).

3. Bispectral and Algebraic Structures

Bispectrality: Wilson polynomials are simultaneous eigenfunctions of a three-term recurrence in $n$ and a second-order difference operator in $x$ with corresponding spectrum (Gisonni et al., 2020, Bergeron et al., 2021).

Rational degenerate Sklyanin algebra: The bispectral operators (second-order S--Heun operators) generate a quadratic algebra, explicitly realized as a rational degeneration of the Sklyanin algebra. The structure incorporates:

• Shift operators $T^+$, $T^-$,
• S--Heun operators $L$, $M_1$, $M_2$, $R_1$, $R_2$,
• Stabilizer subalgebra with commutation relations:

$$[L, M_1]=2L^2, \ [L,M_2]=\{L, M_1\}, \ [M_1, M_2] = \{M_2, L\} - 4L^2$$

where $\{A,B\}=AB+BA$ (Bergeron et al., 2021).

Ladder, contiguity, and structure operators: The S--Heun operators produce parameter-shift ("contiguity") relations for the Wilson polynomials, matching those known from hypergeometric transformations. The four-generator star-algebra closes the full operator structure, with explicit Casimir elements (Bergeron et al., 2021).

Finite-dimensional ("para-Racah") truncation: Imposing $a+b+c+d = -N+1$ annihilates the top-level raising operator, leading to a finite $(N+1)$-dimensional module whose basis elements are (finite) "para-Racah" polynomials, orthogonal on a bi-quadratic lattice (Bergeron et al., 2021).

4. Generating Functions, Connection Relations, and Summations

Classical and generalized generating functions:

• The double Gauss ${}_2F_1$ generating function:

$${}_2F_1(a+ix, b+ix; a+b; \rho)\, {}_2F_1(c-ix, d-ix; c+d; \rho) = \sum_{n=0}^\infty \frac{ \rho^n}{(a+b)_n (c+d)_n n!} W_n(x^2;a,b,c,d)$$

for $|\rho|<1$ (Cohl et al., 2013, Baeder et al., 2014).

• Parameter-shifted generalizations: By inserting the connection relation,

$$W_n(x^2;a,b,c,d) = \sum_{k=0}^n C_{n,k}(a,b,c,d,h) W_k(x^2; a,b,c,h)$$

into these generating functions and rearranging, one obtains new generalized generating function families whose coefficients involve terminating ${}_4F_3$ series in the connection parameter (Cohl et al., 2013, Baeder et al., 2014).

• Further generalizations: The Askey--Wilson generating function generalizes all classical cases and admits additional parameters. Detailed $q$-analogues and further parameter-deformations exist, including nonterminating sums, product transformations, and limiting identities as $q\to 1$ (Cohl et al., 2024, Liu, 2018).

Quadratic and higher transformations: Transformations between basic hypergeometric and hypergeometric generating functions, such as ${}_2\phi_1 \leftrightarrow {}_8\phi_7$ and generalized quadratic transformations, allow migration between parameter sets and enable commutation relations within the Askey scheme (Cohl et al., 2018).

Definite integrals and orthogonality:

Multiplication of generating function identities by the appropriate Wilson polynomial, integration, and use of orthogonality immediately yield new families of definite integrals for hypergeometric functions with weight functions determined by the Wilson measure (Baeder et al., 2014, Cohl et al., 2013).

5. Multivariate and Superintegrable Generalizations

Multivariable Wilson polynomials: Defined by matrix determinants built from univariate Wilson polynomials or their "Theta" building blocks, these generalize the eigenfunction property to $N$-dimensional systems and satisfy a fully factorized norm structure: $$\int_{(0,\infty)^N} \ldots = \delta_{RQ} \prod_{i=1}^N \frac{ h_{R_i+N-i}(a,b,c,d) }{ (a+b+c+d+R_i+N-i-1)_{R_i+N-i}^2 }$$ (Mishnyakov, 2024).

Superintegrable matrix models: The Hermitian eigenvalue model with measure built from the Wilson weight and squared Vandermonde determinant is diagonalized by multivariable Wilson polynomials. Expectation values of symmetric determinant-type functions ("locked" products) admit closed-form evaluations, involving Schur function specializations and hook-content formulas. The corresponding Hamiltonians, arising from higher-rank Wilson difference operators, realize a family of commuting integrals, fulfilling the criterion for classical superintegrability (Mishnyakov, 2024).

6. Applications in Random Matrix Theory, Enumerative Combinatorics, and Integrable Systems

Random matrix theory: The moments and connected correlators of classical unitary ensembles such as the Jacobi Unitary Ensemble (JUE) are expressible in terms of Wilson polynomials. Generating functions for multipoint correlators are given explicitly in terms of matrix-valued expansions whose coefficients are Wilson polynomials, with associated triple-monotone Hurwitz numbers describing the leading-topological expansion (Gisonni et al., 2020).

Enumerative combinatorics: The appearance of Hurwitz numbers, content-product formulas, and Schur polynomial structures in the multivariate theory highlights deep connections with factorizations in the symmetric group, monotonicity constraints, and symmetric function theory (Gisonni et al., 2020, Mishnyakov, 2024).

Integrable systems and bispectral dualities: The algebraic operator structures underpin representation theory and the theory of integrable models (e.g., explicit realization of degenerate rational Sklyanin algebras, S-Heun bispectral operators), with significance for spectral problem formulations and hypergeometric expansions (Bergeron et al., 2021).

7. $q$-Deformations, Jackson–Barnes Integrals, and Special Limits

Askey--Wilson framework: The Askey--Wilson polynomials provide a $q$-deformation, with corresponding $q$-beta integrals, and are governed by twelve-parameter integral identities and double $q$-series transformations. Nonterminating product transformations yield new identities, reducing to Wilson polynomial formulas as $q\to 1$, recapturing all classical properties in this limit (Liu, 2018, Cohl et al., 2024).

Special and limiting cases: Many families within the Askey scheme are recovered by parameter specialization or by sending $q \to 1$, $d \to \infty$, or imposing finite-dimensional truncation ($a+b+c+d=-N+1$), yielding subfamilies such as continuous dual Hahn, para-Racah, and Racah polynomials (Bergeron et al., 2021, Baeder et al., 2014, Cohl et al., 2024).

Summary Table: Selected Structural Aspects of Generalized Wilson Polynomials

Aspect	Classical Wilson	Multivariate Generalizations	$q$-Deformation (Askey–Wilson)
Defining Series	Term. ${}_4F_3$	Determinant of $W_n$	Well-poised $_4\phi_3$
Orthogonality	$\Gamma$-product	Factorized norm, Vandermonde	$q$-beta integral
Recurrence/Difference	3-term / 2nd diff	Raising, Pieri via minors	$q$-recurrence, product formulas
Algebraic Structure	Sklyanin algebra	Higher-rank commutants	$q$-Serre, double $q$-series
Random Matrix Links	One/multipoint	Matrix model moments	$q$-ensemble moment expansions

Generalized Wilson polynomials thus serve as a central object unifying hypergeometric orthogonal families, integrable system symmetries, enumerative combinatorics, and superintegrable matrix models through a web of explicit generating functions, bispectral operators, and multi-variable extensions (Bergeron et al., 2021, Mishnyakov, 2024, Liu, 2018, Cohl et al., 2024, Gisonni et al., 2020, Baeder et al., 2014).