Separating zeros of polynomials using an added interlacing point

Abstract: Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, ${\mathcal{P}_n}$ and ${\mathcal{G}_n}$ can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the $n+1$ zeros of the polynomial $(x-E)\mathcal{P}_n(x)$ interlace with the $k$ zeros of $\mathcal{G}_k$, where $k=n$ or $n+1$. In addition, we show that imposing specific conditions on the extra point $E$ yields full interlacing between the zeros of $\mathcal{P}_n$ and $\mathcal{G}_k$ for a suitable choice of $n$. The approach provides a consolidated framework broadly applicable to both orthogonal and non-orthogonal polynomials and we illustrate this with new interlacing results for zeros of Krawtchouk, Meixner, and Narayana polynomials. We also illustrate that this general approach can be used to recover and refine existing results regarding the complete interlacing of zeros for classical Jacobi and Laguerre polynomials.

• The paper presents a novel framework where the addition of an interlacing point E enables complete zero interlacing of polynomial sequences.
• It rigorously establishes necessary and sufficient conditions through mixed recurrence relations for both classical and nonclassical polynomial families.
• The approach enhances precision in root localization, with implications for spectral analysis, numerical algorithms, and algebraic combinatorics.

Interlacing Zeros of Polynomials via an Added Interlacing Point

Introduction and Problem Formulation

This work presents a systematic approach to understanding when the zeros of distinct polynomial sequences can be made to completely interlace through the introduction of a specific added point. The authors focus on the framework where an auxiliary variable $E$—the “interlacing point”—naturally arises in mixed recurrence relations involving two or more families of polynomials. The primary objects of study are polynomial sequences $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$, and the central goal is to establish conditions under which the zeros of the polynomial $(x-E)\mathcal{P}_{n}(x)$ interlace with those of $\mathcal{G}_{k}(x)$, for $k = n$ or $k = n+1$, as determined by the recurrence structure.

Interlacing of zeros is fundamental in polynomial approximation theory, OSC, and spectral analysis. The extension and generalization of classical interlacing results to both classical orthogonal polynomials (such as Jacobi and Laguerre) and non-orthogonal families (e.g., Narayana) demonstrate the generality of the proposed approach. The analysis leverages general mixed recurrence relations of the form:

$$A(x)\mathcal{P}_{n}(x) = B(x)\mathcal{G}_{j}(x) + H(x)\mathcal{Q}_{k}(x)$$

with specific attention to the case $H(x)=x-E$, so that the additional interlacing point $E$ plays a structural role in controlling the interlacing behavior.

Main Theoretical Results

General Theorems

The central contributions are formalized in several main theorems (Theorems 1, 2*, 2), providing necessary and sufficient conditions for interlacing of zeros between polynomials arising from these mixed recurrence relations:

• Theorem 1: Given appropriate interlacing between $\{\mathcal{P}_n\}$0 and $\{\mathcal{P}_n\}$1, and under mild regularity/hypotheses, the zeros of $\{\mathcal{P}_n\}$2 fully interlace with those of $\{\mathcal{P}_n\}$3 if and only if $\{\mathcal{P}_n\}$4 lies strictly below (or above) all zeros of $\{\mathcal{P}_n\}$5, depending on the direction of interlacing.
• Theorem 2*: For mixed recurrences involving polynomials of degree $\{\mathcal{P}_n\}$6 and $\{\mathcal{P}_n\}$7 in the $\{\mathcal{P}_n\}$8 term, analogous sufficient conditions are determined for when $\{\mathcal{P}_n\}$9 interlaces $\{\mathcal{G}_n\}$0, with the possibility of full interlacing depending on the relative locations of $\{\mathcal{G}_n\}$1 and the zeros.

The primary technical innovation is the acute identification of the role played by the extra point $\{\mathcal{G}_n\}$2, which provides a consolidated and easily verifiable condition for interlacing, in contrast to previous approaches dependent on ad hoc parameter checks specific to classical cases.

Relation to Existing Literature

Included are several direct recoveries and strengthenings of earlier theorems for Jacobi and Laguerre polynomials. In particular, by recasting previous mixed recurrence results in the present framework, the authors provide sharper or necessary-and-sufficient interlacing criteria, where prior results offered only sufficient or partial converse statements.

Applications to Classical and Nonclassical Polynomial Families

The utility of the developed theory is demonstrated through explicit application to several prominent polynomial families. Below is a summary table of the main applications and outcomes.

Family	Mixed Relation Used	Interlacing Property (if $\{\mathcal{G}_n\}$3 satisfies position)	New/Recovered Results
Krawtchouk	Christoffel transform/mixed rec.	$\{\mathcal{G}_n\}$4	New explicit interlacing bounds
Meixner	Christoffel-mixed three term	$\{\mathcal{G}_n\}$5	New explicit interlacing results
Narayana	Specific non-orthogonal sequence	Interlacing between transformed Narayana polynomials	First such result for this nonclassical seq.
Jacobi	Parameter shifts in $\{\mathcal{G}_n\}$6/$\{\mathcal{G}_n\}$7	Various: recovers & extends results in prior work	Strengthens theorems in [Arvesu et al.]
Laguerre	$\{\mathcal{G}_n\}$8 parameter shift	Standard result (Laguerre interlacing criterion)	Recovery and precise criterion for $\{\mathcal{G}_n\}$9

The details for each case include explicit constructions of the recurrence relations, deployment of the general theorems, and the precise location of $(x-E)\mathcal{P}_{n}(x)$0 required for full interlacing.

Technical Insights and Strong Claims

• Unified Framework: The approach unifies disparate results on interlacing into a fully algebraic criterion involving the auxiliary point $(x-E)\mathcal{P}_{n}(x)$1.
• General Applicability: The auxiliary interlacing methodology is independent of orthogonality; it applies to polynomials satisfying mixed-type recursions, e.g., Narayana polynomials.
• Necessity and Sufficiency: The conditions on the position of $(x-E)\mathcal{P}_{n}(x)$2 for full interlacing are both necessary and sufficient in the sense of the given recurrences.
• Completeness of Interlacing: Full interlacing is possible if and only if $(x-E)\mathcal{P}_{n}(x)$3 lies strictly outside the support of zeros for the targeted polynomial sequence, as dictated by the mixed recurrence.
• Counterexamples to Earlier Literature: In specific cases (highlighted in remarks), the paper points out incorrect general claims in previous theorems (e.g., dual placement of outlier zeros for certain shifted Jacobi polynomials), offering corrected analyses and explicit numerical demonstrations.

Implications and Future Directions

The generalization and consolidation achieved open several practical and theoretical avenues:

• Root Localization: The framework enables precise algebraic control over the extremal zeros of related polynomial sequences, which is of significant value for analytic number theory and approximation.
• Extension to Hypergeometric-Type OPS: As the mixed recurrence approach is agnostic to the particular structure (e.g., discrete, continuous, or $(x-E)\mathcal{P}_{n}(x)$4-orthogonality), further classes—such as generalized discrete orthogonal polynomials, Sobolev-type polynomials, and others—can be analyzed within this schema.
• Spectral Theory: Since the zeros of orthogonal polynomials correspond to Gaussian quadrature points, extensions to mixed spectra for non-self-adjoint operators using these results could be developed.
• Algebraic Combinatorics: For non-orthogonal sequences (Narayana, etc.), this methodology provides tools for the investigation of root geometry arising in enumeration and partition polynomials.
• Symbolic/Numeric Algorithms: The explicitness in the position of $(x-E)\mathcal{P}_{n}(x)$5 and its relation to recursive parameters suggests straightforward implementation of symbolic-numeric algorithms for root counting and location, as well as stability analyses for recurrence-based polynomial sequences.

Conclusion

By establishing a general framework for achieving complete interlacing of zeros between polynomial sequences via the introduction of a structurally motivated interlacing point $(x-E)\mathcal{P}_{n}(x)$6, this work both synthesizes and extends previous interlacing results. The general mixed recurrence approach is demonstrated to be widely applicable—enabling new results for Krawtchouk, Meixner, and Narayana polynomials, while simultaneously strengthening classical analyses for Jacobi and Laguerre families. The principal claim is that the precise placement of an auxiliary point $(x-E)\mathcal{P}_{n}(x)$7, as determined by the underlying mixed recurrence, dictates the possibility of full interlacing—a condition that is both necessary and sufficient within this algebraic context (2604.03680).