Zeros and convergent subsequences of Stern polynomials

TL;DR

This paper studies the zeros and convergence properties of polynomial analogues of the Stern diatomic sequence, revealing their distribution in the complex plane and identifying numerous subsequences converging to unique analytic functions.

Contribution

It provides new insights into the zero distribution and convergence behavior of Stern polynomial analogues, extending previous results by Dilcher and Stolarsky.

Findings

Zeros are distributed in the complex plane in a specific pattern.

Uncountably many subsequences converge to unique analytic functions.

Generalizes previous results on Stern polynomials.

Abstract

We investigate Dilcher and Stolarsky's polynomial analogue of the Stern diatomic sequence. Basic information is obtained concerning the distribution of their zeros in the plane. Also, uncountably many subsequences are found which each converge to a unique analytic function on the open unit disk. We thus generalize a result of Dilcher and Stolarsky from their second paper on the subject.