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Separating zeros of polynomials using an added interlacing point

Kerstin Jordaan, Vikash Kumar

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.

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, and can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the zeros of the polynomial interlace with the zeros of , where or . In addition, we show that imposing specific conditions on the extra point yields full interlacing between the zeros of and for a suitable choice of . 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.

Paper Structure

This paper contains 10 sections, 13 theorems, 78 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Interlacing of zeros
  3. Examples
  4. Krawtchouk polynomials
  5. Meixner polynomials
  6. Narayana Polynomials
  7. Jacobi polynomials
  8. Laguerre polynomials
  9. Proofs of the main results
  10. Conclusion

Key Result

Theorem 2.1

Let $\mathcal{G}_{n+1}$ and $\mathcal{Q}_{n+1}$ denote monic polynomials of degree $n+1$ with all real zeros on an (infinite or finite) interval $(a,b)$. Denote the zeros of $\mathcal{G}_{n+1}$ by $\{x_{k,n+1}\}_{k=1}^{n+1}$ and suppose that $\mathcal{G}_{n+1}\prec \mathcal{Q}_{n+1}$ or $\mathcal{Q} where $E\in \mathbb{R}$ and $A(x)>0$ on the interval $(a,b)$. Assume also that $B(E)\neq0$ and that

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.2
  • ...and 15 more