Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.3

Polynomials Whose Coefficients Are Stern Numbers

Karl Dilcher
Department of Mathematics and Statistics
Dalhousie University
Halifax, Nova Scotia B3H 4R2

Larry Ericksen
1212 Forest Drive
Millville, NJ 08332-2512


The main object in this paper is the sequence of polynomials Pn(z) that have Stern numbers as their coefficients; that is, the terms of Stern's diatomic sequence. We derive certain basic properties of these polynomials, investigate the distribution of their real and complex zeros, and prove some results concerning factorizations and resultants. We also consider the (0,1)-polynomials obtained from Pn(z) by taking their coefficients modulo 2. In spite of its simple form, the polynomial sequence is shown to possess some interesting algebraic and analytic properties. Finally, we discuss combinatorial interpretations of the polynomials Pn(z) and indicate ways of generalizing them.

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(Concerned with sequences A000045 A001045 A002487 A011655 A048573 A174868.)

Received July 14 2021; revised version received November 24 2021. Published in Journal of Integer Sequences, November 25 2021.

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