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Keywords:
Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros
Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros
Summary:
The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials $a(n;x)$ defined by $a(0;x)=0$, $a(1;x)=1$, $a(2n;x)=a(n;x^2)$, and $a(2n+1;x)=x\,a(n;x^2)+a(n+1;x^2)$; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that $a(n;x)$ can only have simple zeros, and we state a few conjectures.
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