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Zeros and convergent subsequences of Stern polynomials. (English) Zbl 1271.11027

Let \(\{a(n)\}_{n\geq 0}\) denote Stern’s diatomic sequence. That is, \(a(0)=0\), \(a(1)=1\) and for \(n\geq 1\), \(a(2n)=a(n)\) and \(a(2n+1)=a(n)+a(n+1)\). K. Dilcher and K. Stolarsky [Int. J. Number Theory 3, No. 1, 85–103 (2007; Zbl 1117.11017)] introduced a polynomial analogue of the Stern sequence; the Stern polynomials are defined by \(a(0;x)=0\), \(a(1;x)=1\) and for \(n\geq 1\) by \[ \begin{aligned} a(2n;x)&=a(n;x^2)\\ a(2n+1;x)&=xa(n;x^2)+a(n+1;x^2).\end{aligned} \] In this paper, the author studies the zeros of the Stern polynomials. The author proves that the zeros of the Stern polynomials cluster uniformly near the unit circle in the sense of Weyl. Specifically, the author proves that if \(Z_n(\rho)\) is the number of zeros of \(a(n;x)\) in the annulus \(1-\rho\leq |x|\leq 1/(1-\rho),\) and \(Z_n(\theta_1,\theta_2)\) is the number of zeros of \(a(n;x)\) in the sector \(\theta_1\leq \arg x\leq \theta_2\), then for fixed \(\rho,\theta_1\) and \(\theta_2\) satisfying \(0<\rho<1\) and \(0\leq \theta_1<\theta_2\leq 2\pi\), we have \[ \left|\frac{Z_n(\theta_1,\theta_2)}{\deg a(n;x)}-\frac{\theta_1-\theta_2}{2\pi}\right|<16\sqrt{\frac{3\log n-\log 8}{n}}, \] and \[ 0\leq 1-\frac{Z_n(\rho)}{\deg a(n;x)}<\frac{2}{\rho}\cdot \frac{2\log n-\log 8}{n}, \] for all \(n>0\) which are not powers of \(2\).

MSC:

11B83 Special sequences and polynomials

Citations:

Zbl 1117.11017