Multiple Orthogonal Polynomials on the Unit Circle

• MOPUC are defined via simultaneous orthogonality with respect to multiple measures on the unit circle, using balanced Laurent and Hermite–Padé formulations.
• They exhibit multidimensional Szegő recurrences with nontrivial coefficient structures and determinantal Christoffel–Darboux kernels that underpin spectral analysis.
• Applications include advanced Hermite–Padé approximations, reconstruction of measures via inverse spectral problems, and connections to random matrix theory.

Multiple Orthogonal Polynomials on the Unit Circle (MOPUC) generalize the classical theory of orthogonal polynomials on the unit circle (OPUC) to the setting of simultaneous orthogonality with respect to several measures, possibly complex or matrix-valued, supported on the unit circle $\mathbb{T} = \{z \in \mathbb{C}: |z|=1\}$. The theory encompasses various formulations, including Type I/Type II multiple orthogonality, matrix-valued/mixed orthogonality, generalized Laurent cases, and extensions via Hermite–Padé approximants, all featuring highly nontrivial recurrence structures, determinantal representations, Christoffel–Darboux-type kernels, and rich structural connections to multiple orthogonal polynomials on the real line (MOPRL) via the Szegő mapping and Geronimus-type transformations. Foundational frameworks and results are presented in (Kozhan et al., 2024, Kozhan et al., 8 Jan 2026, Huertas et al., 2024, Kozhan et al., 23 Mar 2026), and (Vaktnäs et al., 2024).

1. Foundations: Definitions and Main Formulations

MOPUC are defined via simultaneous (multiple) orthogonality relations relative to a family of measures $\bm{\mu} = (\mu_1, \ldots, \mu_r)$, each supported on $\mathbb{T}$ and typically of infinite support. There are two main contemporary Laurent-form formulations:

• Balanced Laurent MOPUC (as in (Kozhan et al., 2024)): For multi-index $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$, set $|\bm{n}| = n_1 + \cdots + n_r$. The type II Laurent MOPUC is a Laurent polynomial

$$\Phi_{\bm{n}}(z) = z^{|\bm{n}|/2} + \cdots + K_{-|\bm{n}|/2}z^{-|\bm{n}|/2}$$

satisfying, for $j = 1, \ldots, r$,

$$\oint_{\mathbb{T}} \Phi_{\bm{n}}(z) z^{-k} d\mu_j(z) = 0, \quad k = -n_j/2, \ldots, n_j/2 - 1.$$

• Hermite–Padé/Generalized Laurent MOPUC (as in (Kozhan et al., 8 Jan 2026)): For two multi-indices $\bm{n}, \bm{m} \in \mathbb{N}^r$, consider

$$\Phi_{\bm{n};\bm{m}}(z) = \sum_{k=-|\bm{m}|}^{|\bm{n}|} \kappa_k z^k$$

with the orthogonality conditions

$\bm{\mu} = (\mu_1, \ldots, \mu_r)$0

Special cases $\bm{\mu} = (\mu_1, \ldots, \mu_r)$1 recover the balanced case.

Normality (uniqueness/finiteness of the solution) is determined via nonvanishing of block-Hankel/Toeplitz moment matrices built from the measures’ Fourier coefficients. Angelesco and AT systems of measures admit full normality for all $\bm{\mu} = (\mu_1, \ldots, \mu_r)$2 (Kozhan et al., 2024, Kozhan et al., 23 Mar 2026).

2. Structures: Recurrence Relations and Algebraic Framework

MOPUC admit multidimensional analogues of Szegő recurrences with nontrivial coefficient structure (generalized Verblunsky/Baxter coefficients). For example (Vaktnäs et al., 2024, Kozhan et al., 8 Jan 2026): $\bm{\mu} = (\mu_1, \ldots, \mu_r)$3 with

$\bm{\mu} = (\mu_1, \ldots, \mu_r)$4

and directly or via raising/lowering indices in Hermite–Padé-form (see (Kozhan et al., 8 Jan 2026)) there are coupled nearest-neighbour recurrences connecting all mixed-index polynomials and reversals. These recurrence arrays satisfy discrete zero-curvature/compatibility relations generalizing the Baxter–Szegő and Geronimus identities, and induce algebraic constraints between the recurrence coefficients, such as: $\bm{\mu} = (\mu_1, \ldots, \mu_r)$5 These recurrence structures are central in the stepwise construction of the entire MOPUC lattice and are essential for spectral and asymptotic analysis.

In the matrix-valued or block-orthogonality context, as in the mixed MOLPUC case (Huertas et al., 2024), banded matrix recurrences express multiplication by $\bm{\mu} = (\mu_1, \ldots, \mu_r)$6 in terms of banded actions on step-line-ordered block polynomials, resulting in generalized CMV (Cantero–Moral–Velázquez)-type recurrences and factorizations.

3. Christoffel–Darboux Kernels, Determinantal Formulas, and Biorthogonality

For all MOPUC frameworks, Christoffel–Darboux-type kernels encode the reproducing property and the duality between type I (vector) and type II (scalar or matrix) multiple orthogonal polynomials: $\bm{\mu} = (\mu_1, \ldots, \mu_r)$7 for an admissible path in the multi-index lattice, and satisfy reproducing identities of the form

$\bm{\mu} = (\mu_1, \ldots, \mu_r)$8

Explicit determinantal representations (of Heine–type) for the polynomials and the recurrence coefficients are available via block-moment matrices, enabling direct access to the coefficients from the data (see (Kozhan et al., 8 Jan 2026, Huertas et al., 2024)).

Mixed and matrix-valued cases (Huertas et al., 2024) require careful handling of multi-band and biorthogonal systems, with Christoffel–Darboux kernels expressed in terms of banded matrix truncations and quasi-determinants, integral identities, and explicit connection formulas between perturbed and unperturbed polynomial systems under Christoffel/Geronimus transformations.

4. Zero Distributions, Normality, and Szegő Mapping

A key structural theorem is the zero localization for MOPUC under Angelesco and AT systems: all zeros of the type II polynomials are strictly inside the open unit disk $\bm{\mu} = (\mu_1, \ldots, \mu_r)$9 [(Kozhan et al., 23 Mar 2026), Theorem 6.1 of (Kozhan et al., 2024)]. The proof employs para-orthogonal polynomials, Blaschke product techniques, and Chebyshev system arguments; the zeros of para-orthogonal analogues always lie on the circle, while those of the type II MOPUC are strictly confined to $\mathbb{T}$0. This is an exact analogue of real-line multiple orthogonality theory but transposed to $\mathbb{T}$1.

Via the Szegő mapping ($\mathbb{T}$2), a bijective correspondence is established between certain families of Laurent MOPUC and multiple orthogonal polynomials on $\mathbb{T}$3, along with explicit transformation formulas and Geronimus relations connecting recurrence coefficients (Kozhan et al., 8 Jan 2026, Kozhan et al., 2024). Explicitly, for normal diagonal indices: $\mathbb{T}$4

The zero location theorems ensure control over the limiting distribution of roots, which, in explicit hypergeometric cases (see (Wolfs, 2024)), suggests condensation on analytic arcs, determined by free multiplicative convolution and potential theory analogues.

5. Mixed, Matrix, and Block Orthogonality: MOLPUC Framework

Beyond scalar and vector MOPUC, the MOLPUC framework (Huertas et al., 2024) develops a matrix-valued orthogonality setting: for $\mathbb{T}$5 measures $\mathbb{T}$6 on $\mathbb{T}$7, one constructs families of matrix-valued Laurent polynomials using Gauss–Borel factorization of moment matrices with CMV-type monomial ordering:

• Type II (matrix-valued) polynomials $\mathbb{T}$8
• Type I (matrix-valued) polynomials $\mathbb{T}$9 with $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$0 block-triangular. The resulting families are biorthogonal relative to $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$1 and each other, with explicit degree, recurrence, and Christoffel–Darboux relationships. Factorizations induce banded recurrences generalizing both classical CMV and higher-band Szegő-type operators. This machinery directly supports Christoffel and Geronimus spectral transformations and their quasi-determinantal characterizations of the perturbed systems.

6. Applications and Directions: Hermite–Padé, Asymptotics, and Integrable Models

MOPUC feature as solutions to generalized Hermite–Padé approximation problems (two-point Hermite–Padé), with equivalence between approximant characterization and orthogonality (Kozhan et al., 8 Jan 2026, Kozhan et al., 2024, Vaktnäs et al., 2024). Their kernels and recurrence arrays connect to prediction theory for multivariate stationary processes on the unit circle, generalized quadrature schemes, and, through conjectural links, to random-matrix ensembles with multi-logarithmic interaction potentials (Wolfs, 2024, Kozhan et al., 2024).

Explicit constructions with hypergeometric weights on the unit circle (Bessel-type case) yield MOPUC with Gamma-function moments, Mellin-transform techniques, and explicit Rodrigues formulas (Wolfs, 2024). These settings allow for the functional calculation of moments, computation of explicit kernel formulas, and conjectural asymptotic analysis via free convolution and S-transform techniques.

Further connections, open questions, and future research avenues include:

• Transfer of Riemann–Hilbert asymptotic methods and integrable probability techniques from MOPRL to the circle case
• Full classification of normality for general measure configurations
• Inverse spectral problems: reconstructing measures on $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$2 from arrays of recurrence data
• The spectral and zero distribution analysis of associated CMV-block operators and their links to matrix-valued Jacobi operators on $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$3
• Applications to random polynomial ensembles and noncolliding path models on the circle, as proposed in (Wolfs, 2024, Vaktnäs et al., 2024).

7. Comparison with Real-Line and Classical OPUC Theory

MOPUC exhibit close structural parallels with MOPRL but with distinctive analytic and algebraic features:

• The unit circle framework replaces three-term $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$4 recurrences with higher-order, multidimensional Szegő recurrences mixing polynomials and reversals;
• The Christoffel–Darboux and Geronimus transforms retain their essential roles but are formulated in terms of Laurent polynomial spans and CMV step-line orderings;
• Angelesco and AT systems provide the classification of full normality and guarantee classical zero localization and determinantal positivity on $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$5;
• Para-orthogonal polynomials and Blaschke-product arguments give matrix- and infinitely-band generalizations of the scalar OPUC root theory.

For $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$6, all the frameworks collapse to the classical Szegő-Baxter theory of OPUC, and the multidimensional recurrence/compatibility structures reduce to the one-parameter Verblunsky/Baxter arrays. For $\bm{n} = (n_1, \ldots, n_r) \in \mathbb{N}^r$7, the essential phenomena of multiple orthogonality, compatibility, and spectral transformation become substantially richer and more intricate (Kozhan et al., 2024, Kozhan et al., 8 Jan 2026, Kozhan et al., 23 Mar 2026, Huertas et al., 2024, Vaktnäs et al., 2024).