A partial-sum deformation for a family of orthogonal polynomials

TL;DR

The work introduces and analyzes partial-sum deformations of orthogonal polynomials via $q_m(x;t)=\sum_{n=0}^m t^n p_n(x)$, framing them through generating functions and a four-term recurrence. It proves that zeros are real for large $t$ and examines their evolution as $t$ varies, including a conjectured single critical value where real zeros give way to complex pairs. Through Hermite, Charlier, and Lommel examples, the paper reveals diverse zero-trajectory phenomena and links the small-$t$ limit to Szegő curves, hinting at deep structural connections beyond classical orthogonality. The results open avenues for further study of deformations of orthogonal polynomials and their zero distributions, with potential ties to matrix-valued OPS and heat-equation analogies.

Abstract

There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)=\sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal polynomials $\{p_n(x)\}_{n\ge0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.

Figures (5)

• Figure 4.1: Zeros of the partial sums of Hermite polynomials $q_m(x;t)$ for $m=100$; $t=20$ (left), $t=3.48$ (center) and $t=1.69$ (right).
• Figure 4.2: Left: Zeros of rescaled partial sums of Hermite polynomials. The small circles in the complex plane are the zeros of $\widetilde{q}_m(x;t_{\max})$. The blue and pink lines are the trajectories of these zeros in the range $t_{\min}$ to $t_{\max}$. The large circles on the real line are the zeros of the classical Hermite polynomials $H_m(\sqrt{m} \, x)$. In this example: $m=10$, $t_{\max}=6$, $t_{\min}=0.1$. Right: Zeros of partial sums of rescaled Charlier polynomials $q^{(3)}_{m}(m x;t)$ from $t_{\min}$ to $t_{\max}$. The large circles on the real line are the zeros of the classical Charlier polynomials $C_{m}^{(3)}(m x)$. In this example: $m=10$, $t_{\max}=6$, $t_{\min}=0.57$.
• Figure 4.3: Zeros of the partial sums of Lommel polynomials $q^{(3)}_m(x;t)$ for $m=100$; $t=1.8$ (left), $t=0.286$ (center) and $t=0.01$ (right).
• Figure 4.4: Zeros of the partial sums of Hermite polynomials $q_m(x;t)$ and the scaled Szegő curve $|ze^{1-z}|=1$ where $z=2xt/m$ for $t=0.00001$ and $m=40$ (left), $m=70$ (right).
• Figure 4.5: Zeros of the partial sums of Charlier polynomials $q^{(a)}_m(x;t)$ and the scaled Szegő curve $|ze^{1-z}|=1$ where $z=-xt/am$ for $a=3$, $t=0.000001$ and $m=40$ (left), $m=70$ (right).

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Proposition 2.2
• Corollary 2.3
• proof : Proof of Proposition \ref{['prop:gengenfunctionasdetofalmosttridiagmatrix']}
• Remark 2.4
• Corollary 2.5
• Lemma 2.6
• proof
• Corollary 2.7