Geronimus Relations in Orthogonal Polynomials

• Geronimus relations are canonical transformation formulas that modify orthogonal polynomial systems via rational spectral changes and mass augmentations.
• They provide explicit connection formulas, modified recurrences, and determinantal structures derived from Cauchy transforms and moment functionals.
• Applications span scalar, matrix, and multivariate cases, underpinning integrable systems and asymptotic spectral analysis of orthogonal polynomials.

A Geronimus relation is a canonical transformation formula governing the effect of a rational spectral transformation—division by a linear or higher-degree polynomial, potentially augmented with mass terms at the zeros—on orthogonal polynomial systems (OPS), their recurrences, and their associated spectral data. This family of relations underlies the analytic, algebraic, and spectral structure of rational modifications of measures, moment functionals, bilinear forms, and their associated OPS, both in the scalar and matrix/multivariate settings. The classical Geronimus relation connects the new and old orthogonal polynomials by a succinct rank-one or rank-two relation whose coefficients are determined explicitly via Cauchy transforms, determinants, or moment functionals.

1. Canonical Formulation of the Geronimus Transformation

Let $u$ be a quasi-definite scalar or matrix-valued moment functional, and $a$ a shift point not in the support of $u$. The elementary scalar Geronimus transformation is defined via

$$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$

where $\delta$ is the Dirac delta functional and $\tau$ a mass parameter. In the context of OPs, let $(P_n)$ denote the original monic OPS. The Geronimus-transformed $\check P_n$ satisfies the rank-one connection

$\check P_n(x) = P_n(x) + A_n\,P_{n-1}(x)$

where the connection coefficient $A_n$ is given explicitly by ratio-of-determinant or Cauchy-integral formulas (Ariznabarreta et al., 2015). In matrix, Laurent, or multivariate extensions, the rational factor may be of higher (even vector) degree, with masses and distributions supported at each spectral zero (Álvarez-Fernández et al., 2016, Mañas et al., 2024, Ariznabarreta et al., 2015).

2. Connection Formulas and Modified Recurrences

A salient feature is that the Geronimus relation induces an explicit connection between the perturbed and unperturbed OPS. For the classical Laguerre measure perturbed at $a$0,

$a$1

with

$a$2

where $a$3 is the function of the second kind and $a$4 (Deaño et al., 2015).

The corresponding three-term recurrence for the Geronimus-perturbed OPS becomes

$a$5

with

$a$6

where $a$7 and $a$8 are the diagonal/off-diagonal recurrences of the original OPS (Deaño et al., 2015). Analogous formulas exist for matrix-valued, multivariate, and Laurent cases (Álvarez-Fernández et al., 2016, Ariznabarreta et al., 2015, Ariznabarreta et al., 2016).

3. Determinantal and Quasideterminantal Structures

Geronimus relations universally yield determinantal or, in noncommutative cases, quasi-determinantal formulas for the perturbed polynomials. In the scalar case, for a polynomial $a$9 (multiple Geronimus transformation),

$u$0

and for $u$1,

$u$2

encoding the dependence on kernel evaluations and derivative data at the mass points (Derevyagin et al., 2014, Derevyagin et al., 2013).

For matrix perturbations, perturbed type II and type I polynomials are given in terms of block determinants or, equivalently, Gel'fand–Retakh quasideterminants in the multivariate or multiparameter case (Ariznabarreta et al., 2015, Álvarez-Fernández et al., 2016, Mañas et al., 2024): $u$3 where $u$4 denotes the spectral jet at the zeros of the perturbing polynomial.

4. Sobolev Inner Products and Multiple Geronimus Transformations

Successive application of Geronimus steps or higher-order transformations naturally yields generalized inner products, including discrete Sobolev types. For instance, the double Geronimus transform leads to (Derevyagin et al., 2013, Derevyagin et al., 2014): $u$5 which can be recast as a non-diagonal Sobolev inner product,

$u$6

The corresponding orthogonal families satisfy higher-bandwidth (pentadiagonal, etc.) recurrences whose coefficients are given explicitly by the Geronimus data.

Every discrete Sobolev inner product corresponds to a unique multiple Geronimus transformation; see (Derevyagin et al., 2014) for the equivalence theorem.

5. Spectral and Integrable/Algebraic Aspects

The Geronimus transformation realizes a right inverse to the Christoffel (multiplicative) spectral transform and admits a Darboux factorization of the underlying Jacobi or Hessenberg (block) matrices. Specifically, for each step, the shifted Jacobi matrix $u$7 factorizes as $u$8, with the perturbed matrix $u$9 recovering the new recurrence data (García-Ardila et al., 2021, Rolanía et al., 2019). In the multivariate and block settings, Geronimus maps correspond to UL factorization of the block-moment (Hankel or Toeplitz) matrices (Ariznabarreta et al., 2015, Álvarez-Fernández et al., 2016).

In integrable systems, Geronimus transformations induce explicit shifts of Baker and wave functions in multispectral 2D Toda lattice hierarchies, and correspond to vertex-operator insertion in KP/Toda tau-functions (Ariznabarreta et al., 2015). The resolvent and connector matrices controlling the transformation appear block-banded, preserving integrability of the hierarchy.

6. Geronimus Relations for Laurent, Multiple, and Matrix Orthogonality

Geronimus relations extend naturally to Laurent polynomials, multiple orthogonal polynomials (MOPs), and matrices. On the unit circle, the classical Szegő mapping pairs with the Geronimus relation to transfer Verblunsky coefficients into Jacobi parameters, now generalized to multilevel block/compatibility recurrences in MOPs (Kozhan et al., 8 Jan 2026, Kozhan et al., 23 Mar 2026).

For multiple orthogonality on the unit circle, the generalized Geronimus relations express the real-line recurrence coefficients $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$0, $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$1 in terms of the MOPUC Verblunsky data and cross-compatibility constants, capturing the structural transfer of nearest-neighbour recurrences between circle and real-line (Kozhan et al., 8 Jan 2026, Kozhan et al., 23 Mar 2026).

Matrix variants involve transformations by division with a matrix polynomial and addition of mass matrices (distributions supported at Jordan chains), with explicit block-determinantal connection formulas and spectral mapping of the Markov–Stieltjes matrix function via linear-fractional transformations (Mañas et al., 2024, Álvarez-Fernández et al., 2016).

7. Applications, Asymptotic and Spectral Consequences

Geronimus relations yield explicit strong and relative asymptotics for the perturbed OPs, as in the Laguerre case, with precise asymptotics for the connection coefficients $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$2 and the new polynomials in all classical regimes (outer, inner, Mehler–Heine) (Deaño et al., 2015). In the theory of zeros, the main effect is the “pulling” of exactly one zero toward each mass/shifting point as the corresponding mass grows, while the rest of the zeros interlace or converge to the spectrum of an auxiliary measure (Branquinho et al., 2014).

Additional consequences include the isospectrality (up to added discrete points) and transfer of ratio asymptotics, spectral invariance or semi-invariance of Nevai classes, and the derivation of higher-order difference or differential equations (ladder operators and holonomic ODEs) satisfied by the Geronimus-perturbed systems (Branquinho et al., 2014, Kumar et al., 2024).

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Representative Table: Classical Geronimus Relation (Scalar Case, Mass $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$3 at $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$4)

Feature	Formula (example)	Reference
Measure	$$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$5	(Deaño et al., 2015)
OP Connection	$$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$6	(Deaño et al., 2015)
Recurrence Coeffs	$$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$7, $$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$8	(Deaño et al., 2015)
Cauchy/2nd-kind func.	$$(x - a)\,\check u = u \quad \Longleftrightarrow \quad \check u = \frac{u}{x - a} + \tau\,\delta(x - a)$$9	(Deaño et al., 2015)
$\delta$0 (Laguerre)	$\delta$1 (see main text above)	(Deaño et al., 2015)

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• (Deaño et al., 2015): Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure
• (Ariznabarreta et al., 2015): Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• (Derevyagin et al., 2013): A note on the Geronimus transformation and Sobolev orthogonal polynomials
• (Derevyagin et al., 2014): Multiple Geronimus transformations
• (Álvarez-Fernández et al., 2016): Transformation theory and Christoffel formulas for matrix biorthogonal polynomials on the real line
• (Ariznabarreta et al., 2016): CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• (Mañas et al., 2024): General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials
• (García-Ardila et al., 2021): Associated orthogonal polynomials of the first kind and Darboux transformations
• (Branquinho et al., 2014): Zeros of orthogonal polynomials generated by the Geronimus perturbation of measures
• (Rolanía et al., 2019): Geronimus transformations for sequences of $\delta$2-orthogonal polynomials
• (Kozhan et al., 8 Jan 2026, Kozhan et al., 23 Mar 2026): Szegő Mapping and Hermite–Padé Polynomials for Multiple Orthogonality on the Unit Circle, Zeros of Laurent multiple orthogonal polynomials on the unit circle
• (Kumar et al., 2024): Recovering orthogonality from quasi-nature of Spectral transformations

Geronimus relations thus form a central component in the modern transformation theory of (multiple, matrix, discrete, and Laurent) orthogonal polynomials, encoding analytic, algebraic, and spectral data under rational, mass-augmented modifications.