A polynomial analogue to the Stern sequence. (English) Zbl 1117.11017
The Stern sequence \(a(n)\) here is defined by \(a(2n)=a(n)\) and \(a(2n+1)=a(n)+a(n+1)\) for \(n\geq1\) with the initial conditions \(a(0)=0\) and \(a(1)=1\). The study of this sequence has a long history, which is briefly reviewed in the introduction. The authors consider a natural extension to the Stern polynomials defined by \(a(2n;x)=a(n;x^2)\) and \(a(2n+1;x)=xa(n;x^2)+a(n+1;x^2)\), which turn out to exhibit several interesting properties. In particular, they are closely connected to Stirling polynomials (of the second kind) and Chebyshev polynomials.
Reviewer: Hsien-Kuei Hwang (Taipei)
MSC:
| 11B83 | Special sequences and polynomials |
| 11B37 | Recurrences |
| 11B75 | Other combinatorial number theory |
| 11B50 | Sequences (mod \(m\)) |
Online Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,k) is the coefficient of t^k in the Stern polynomial B(n,t) (n>=0, k>=0).References:
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