Interlacing of zeros of polynomials completed with two additional points

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 $P_n^{(α,β)}$ and $P_{n+1}^{(α+1,β+1)}$. 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.

• The paper constructs an explicit quadratic polynomial to determine the two additional points that complete the interlacing pattern between polynomial zeros.
• It employs a mixed recurrence relation and discriminant analysis to characterize admissible configurations for Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi families.
• Findings have practical implications for Gaussian quadrature and spectral methods by ensuring node separation and numerical stability.

Interlacing of Zeros in Polynomials Augmented by Two Additional Points

Introduction and Motivations

This paper investigates the interlacing properties of zeros in pairs of polynomial sequences, especially in cases where a canonical interlacing fails by precisely two points. Such scenarios frequently arise in sequences of orthogonal polynomials under parameter shifts or degree changes that disrupt the classical interlacing structure. The central objective is to ascertain explicit algebraic constructions (typically quadratic polynomials) whose zeros, when adjoined to a non-interlacing configuration, restore full interlacing with a related polynomial sequence.

The approach leverages a general mixed recurrence relation involving quadratic perturbations, yielding a uniform algebraic framework to characterize and locate the additional interlacing points. The authors apply this construction to several classical families: Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi polynomials. The results improve the explicitness and scope of known interlacing properties, resolving open questions on parameter-shifted families.

General Theory: Completion of Interlacing by Two Points

The core technical apparatus is a general mixed recurrence:

$$A(x)\mathcal{P}_n(x) = B(x)\mathcal{G}_{n+1}(x) \mp (x-E_1)(x-E_2)\mathcal{Q}_n(x),$$

where $\mathcal{G}_{n+1}$, $\mathcal{Q}_n$, and $\mathcal{P}_n$ denote monic polynomials of degree $n+1$, $n$, and $n$, respectively, with all zeros real and simple. The positions of the additional points $E_1$ and $E_2$ are algebraically determined as roots of a quadratic representation derived from the recurrence and the structure of the polynomial family.

The fundamental theorems (Theorems 1 and 2) provide necessary and sufficient conditions on $E_1, E_2$ for the zeros of $\mathcal{G}_{n+1}$0 and $\mathcal{G}_{n+1}$1 to be fully interlacing. The configurations under which $\mathcal{G}_{n+1}$2 and $\mathcal{G}_{n+1}$3 suffice for complete interlacing (as opposed to only one of the two points being required in special subintervals) are exhaustively described, with sharp results on when such configurations are impossible.

A key insight is that the two exceptional points cannot both lie within any subinterval formed by two consecutive zeros of $\mathcal{G}_{n+1}$4 or both outside the support. The authors present combinatorial and algebraic arguments to categorize all admissible configurations.

Applications to Classical Orthogonal Families

Jacobi Polynomials

For Jacobi polynomials $\mathcal{G}_{n+1}$5 and their parameter-upshifted counterparts $\mathcal{G}_{n+1}$6, the authors derive explicit expressions for the two interlacing-completing points via a quadratic factor constructed from a mixed three-term recurrence. This calculation improves on prior results [Arvesu, Driver, Littlejohn, Ramanujan J. 2023], resolving ambiguities in the relative positions of four zeros that could previously only be approached via numerical experimentation.

In particular, when the parameters and degree satisfy $\mathcal{G}_{n+1}$7 and $\mathcal{G}_{n+1}$8 (with $\mathcal{G}_{n+1}$9 the zeros of $\mathcal{Q}_n$0), the zeros of $\mathcal{Q}_n$1 and $\mathcal{Q}_n$2 interlace in the strong sense: every root of one polynomial lies between roots of the other. Numerical evidence supports the generality and sharpness of these results.

Meixner-Pollaczek Polynomials

For Meixner-Pollaczek polynomials, the parameter shift also breaks canonical interlacing. The authors derive a real quadratic whose roots depend explicitly on $\mathcal{Q}_n$3, $\mathcal{Q}_n$4, and $\mathcal{Q}_n$5 in the model polynomial $\mathcal{Q}_n$6. The discriminant analysis yields a precise admissible range for $\mathcal{Q}_n$7 so that the two points are real and allow for restored interlacing.

This directly addresses an open question in [Jooste, Jordaan, Numer. Algorithms 2025], giving not only partial interlacing conditions but also explicit criteria for strong (complete) interlacing under parameter constraints.

Pseudo-Jacobi Polynomials

Pseudo-Jacobi polynomials, defined in terms of hypergeometric functions and parameterized by $\mathcal{Q}_n$8 and $\mathcal{Q}_n$9, present an analogous problem for finite sequences. A detailed discriminant computation for the auxiliary quadratic polynomial in $\mathcal{P}_n$0 gives conditions on $\mathcal{P}_n$1 and $\mathcal{P}_n$2 under which two real points suffice to complete interlacing. Again, the result is new for this family, given the lack of simple explicit relations for their zeros.

Numerical Results and Conjectures

Extensive numerical experiments reinforce that the interlacing structures established are not simply necessary but also essentially sufficient: the zeros of the original and parameter-shifted polynomials consistently arrange themselves to satisfy the completed interlacing, provided the quadratic’s roots lie in prescribed positions relative to the support.

The authors pose two conjectures (for Meixner-Pollaczek and Pseudo-Jacobi polynomials), essentially asserting that for any configuration where $\mathcal{P}_n$3 and $\mathcal{P}_n$4 lie in different subintervals between zeros of $\mathcal{P}_n$5, then the refined interlacing (with one exceptional zero in each sub-interval) always holds. These statements, if proven generally, would unify the theory further.

Theoretical and Practical Implications

On the theoretical side, this work generalizes classical separation (Stieltjes) theorems, providing constructive algebraic tools to deal with situations where standard interlacing breaks down. The findings clarify which modifications (parameter shifts, explicit completion points) restore a “full” interlacing pattern.

From a practical (computational) viewpoint, the structure of the interlacing zeros is essential in areas such as Gaussian quadrature, spectral methods, and rational interpolation, as the placement of nodes directly impacts accuracy and convergence properties. These results provide new tools for constructing sets of nodes with guaranteed separation—even when orthogonality, and thus classical interlacing, is not preserved by parameter changes.

Furthermore, the explicit nature of the degree-two auxiliary polynomials enables easy computation of the critical points and direct verification of interlacing—a substantial improvement over approaches that rely on implicit separation or numerically delicate root-finding in complex parameter regimes.

This methodology also opens the way for further research into completion of interlacing in more general polynomial families, including non-orthogonal or $\mathcal{P}_n$6-orthogonal cases, albeit with significantly increased algebraic complexity for higher-degree completion.

Conclusion

The paper provides a unified algebraic framework for completing the interlacing of zeros of polynomial sequences using two explicit additional points, with guaranteed sharpness and explicit location formulas for several important classical families. Theoretical results are supported by extensive verification, and the analysis resolves a number of open problems and ambiguities in the existing literature. The general theory has broad implications for the study of polynomial zeros and construction of separated node sets, and suggests further avenues for the explicit algebraic control of zero configurations in parameterized polynomial families.

Reference: "Interlacing of zeros of polynomials completed with two additional points" (2604.25692).