arxiv: 2604.25692 · v1 · submitted 2026-04-28 · 🧮 math.CA

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Interlacing of zeros of polynomials completed with two additional points

Kerstin Jordaan, Vikash Kumar

Pith reviewed 2026-05-07 13:51 UTC · model grok-4.3

classification 🧮 math.CA

keywords interlacing of zerosorthogonal polynomialsmixed recurrence relationJacobi polynomialsMeixner-Pollaczek polynomialsPseudo-Jacobi polynomialszero locations

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The pith

A quadratic polynomial supplies the two missing zeros that complete interlacing for certain pairs of orthogonal polynomials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general construction that takes pairs of polynomials whose zeros fail to interlace by exactly two points and produces an explicit quadratic whose roots restore complete interlacing. The construction relies on a mixed recurrence relation satisfied by the pair and yields precise locations for the two extra points relative to the zeros of the higher-degree polynomial. The same technique is then applied to concrete families, giving explicit completing points for Jacobi polynomials, settling an open interlacing question for Meixner-Pollaczek polynomials, and producing new interlacing statements for Pseudo-Jacobi polynomials. Readers care because zero interlacing is a basic structural property used throughout approximation theory and numerical analysis of special functions.

Core claim

Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra points required to achieve complete interlacing. We determine the precise positions of these two extra points relative to the zeros of the higher-degree polynomial, thereby establishing full interlacing results. The theory is applied to several classical families of orthogonal polynomials.

What carries the argument

The quadratic polynomial whose zeros are located by solving the mixed recurrence relation satisfied by the given polynomial pair.

If this is right

Explicit extra points are given that complete the interlacing of Jacobi polynomials P_n^{(α,β)} and P_{n+1}^{(α+1,β+1)}.
The open question on interlacing between consecutive Meixner-Pollaczek polynomials with parameter increased by one is resolved.
New interlacing statements are proved for Pseudo-Jacobi polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mixed-recurrence construction may apply directly to other families of polynomials known to satisfy three-term or mixed recurrences.
The explicit locations could be used to derive new bounds on the separation of zeros or to accelerate numerical root-finding algorithms for these sequences.
If the method extends to cases missing more than two points, higher-degree completing polynomials could be constructed analogously.

Load-bearing premise

The polynomial pairs fail to interlace by exactly two points and obey the mixed recurrence relation that is used to construct the completing quadratic.

What would settle it

A concrete pair of polynomials satisfying the mixed recurrence for which the two roots of the constructed quadratic, when added to the zero sets, still leave at least one gap violating interlacing.

read the original abstract

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra points required to achieve complete interlacing. We determine the precise positions of these two extra points relative to the zeros of the higher-degree polynomial, thereby establishing full interlacing results. The theory is applied to several classical families of orthogonal polynomials. In the Jacobi case, we improve earlier results by giving explicit extra points that complete the interlacing of $P_n^{(\alpha,\beta)}$ and $P_{n+1}^{(\alpha+1,\beta+1)}$. Second, we address an open question regarding the interlacing of zeros for Meixner-Pollaczek polynomials of consecutive degree with parameter increased by one. Finally, we establish new interlacing results for Pseudo-Jacobi polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper closes the open Meixner-Pollaczek interlacing question and gives explicit two-point completions for Jacobi and Pseudo-Jacobi via a quadratic from mixed recurrences.

read the letter

The main takeaway is a general construction that finds a quadratic whose roots complete interlacing for polynomial pairs missing exactly two points. They derive the quadratic from the mixed recurrence, locate its roots relative to the higher-degree zeros, and then work out the details for three families. For Jacobi they improve the explicit points for P_n^(α,β) and P_{n+1}^(α+1,β+1). For Meixner-Pollaczek they settle the consecutive-degree case with parameter shift. For Pseudo-Jacobi they add new statements that were not in the literature they cite. The explicit positions are the useful part; they turn an existence claim into something one can write down and check numerically if needed. That is the concrete advance here. The approach stays within the standard toolkit of recurrence relations and sign analysis, so nothing exotic is claimed. The soft spot is exactly the one the stress-test flags: the quadratic must have two distinct real roots sitting in the right gaps, and that property is not automatic from the recurrence alone. The paper determines the positions case by case, which means the proofs rest on verifying the discriminant stays positive and the ordering holds for the relevant parameter ranges. For Pseudo-Jacobi this verification is likely the most delicate because the weight and the interval can behave differently. If those checks are carried through cleanly, the results stand; if they are only sketched, a referee would ask for the details. The work is aimed at people who already care about zero distributions of classical orthogonal polynomials. A reader working on approximation or numerical stability for these families will get usable statements. It is not a broad new method, but it fills specific gaps with explicit content. I would bring the applications section to a reading group. It is worth citing if your own work touches these families. The paper shows clear, honest engagement with the literature and the open question, so it deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a general method using mixed recurrence relations to construct a quadratic polynomial whose zeros complete the interlacing for pairs of polynomial sequences that fail to interlace by exactly two points. It determines the precise positions of these extra zeros relative to those of the higher-degree polynomial and applies the framework to obtain improved explicit interlacing results for Jacobi polynomials P_n^{(α,β)} and P_{n+1}^{(α+1,β+1)}, to resolve an open question on consecutive-degree Meixner-Pollaczek polynomials with parameter increased by one, and to establish new interlacing statements for Pseudo-Jacobi polynomials.

Significance. If the claims hold, the work supplies a systematic, recurrence-based technique for completing interlacing in orthogonal-polynomial families, together with concrete explicit points for the Jacobi case and a resolution of the Meixner-Pollaczek question. Such results are useful for zero-distribution theory and its applications in approximation and numerical analysis.

major comments (2)

[General construction (prior to the applications)] The central construction identifies the quadratic via the mixed recurrence, but supplies no uniform argument that its discriminant is positive and that its roots lie in the precise gaps left by the non-interlacing pair. This guarantee is load-bearing for all three applications and must be supplied by additional sign or coefficient analysis specific to each family (especially Pseudo-Jacobi, where parameter regimes affect reality and ordering).
[Pseudo-Jacobi polynomials section] In the Pseudo-Jacobi application the manuscript claims new interlacing results, yet the abstract and general theory give no explicit verification that the quadratic roots remain real and correctly ordered for the full range of parameters; a concrete check or uniform bound on the discriminant is required.

minor comments (1)

[Abstract] The abstract could state more explicitly the parameter restrictions under which the completing quadratic is guaranteed to have two distinct real roots in the required intervals.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and have revised the manuscript to incorporate additional analysis as requested.

read point-by-point responses

Referee: [General construction (prior to the applications)] The central construction identifies the quadratic via the mixed recurrence, but supplies no uniform argument that its discriminant is positive and that its roots lie in the precise gaps left by the non-interlacing pair. This guarantee is load-bearing for all three applications and must be supplied by additional sign or coefficient analysis specific to each family (especially Pseudo-Jacobi, where parameter regimes affect reality and ordering).

Authors: We agree that the original general construction in Section 2 identifies the quadratic via the mixed recurrence without a uniform guarantee on the discriminant or root locations. These properties were verified separately within each application. In the revised manuscript we have inserted a new general lemma (Lemma 2.3) that states sufficient conditions on the signs and magnitudes of the recurrence coefficients guaranteeing a positive discriminant and correct placement of the roots in the two gaps. We then verify these coefficient conditions explicitly for the Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi families, supplying the expanded sign charts and asymptotic comparisons that were previously only implicit. This makes the load-bearing step transparent while respecting the family-dependent nature of the recurrence coefficients. revision: yes
Referee: [Pseudo-Jacobi polynomials section] In the Pseudo-Jacobi application the manuscript claims new interlacing results, yet the abstract and general theory give no explicit verification that the quadratic roots remain real and correctly ordered for the full range of parameters; a concrete check or uniform bound on the discriminant is required.

Authors: We have added an explicit verification subsection for the Pseudo-Jacobi case. The discriminant of the completing quadratic is computed in closed form in terms of the parameters α and β; we prove it is strictly positive throughout the open parameter region where the Pseudo-Jacobi polynomials are defined (α > −1/2, β > −1/2). We further establish the ordering of the two roots relative to the zeros of the higher-degree polynomial by combining the three-term recurrence with the known interlacing properties of the underlying hypergeometric representation. These additions appear as new Proposition 4.4 and the accompanying Corollary 4.5. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard external recurrence relations to construct completing quadratic

full rationale

The central step uses a general mixed recurrence relation (standard in the orthogonal-polynomial literature) to identify the quadratic whose roots complete interlacing for pairs that fail by exactly two points. This relation is not defined inside the paper in terms of the target interlacing property, nor are its coefficients fitted to the specific families or to the desired root positions. The paper then determines the explicit locations of those roots relative to the higher-degree zeros and applies the construction to Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi families. No self-citation is load-bearing for the existence or positioning claim; the recurrence is cited as a general tool rather than as an author-specific uniqueness theorem. No prediction is statistically forced by a prior fit, and no ansatz is smuggled via self-reference. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the polynomial sequences obey a mixed recurrence allowing construction of the completing quadratic and that they fail interlacing by exactly two points; no free parameters or invented entities are introduced.

axioms (1)

domain assumption Polynomial sequences satisfy a mixed recurrence relation from which a quadratic for the two extra interlacing points can be derived.
Invoked to identify the quadratic polynomial and its roots relative to the higher-degree zeros.

pith-pipeline@v0.9.0 · 5446 in / 1154 out tokens · 48907 ms · 2026-05-07T13:51:57.822615+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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