A Polynomial Analogue to the Stern Sequence

Abstract

We extend the Stern sequence, sometimes also called Stern's diatomic sequence, to polynomials with coefficients 0 and 1 and derive various properties, including a generating function. A simple iteration for quotients of consecutive terms of the Stern sequence, recently obtained by Moshe Newman, is extended to this polynomial sequence. Finally we establish connections with Stirling numbers and Chebyshev polynomials, extending some results of Carlitz. In the process we also obtain some new results and new proofs for the classical Stern sequence.

The Moments of Minkowski Question Mark Function: The Dyadic Period Function

“…This series is absolutely and uniformly convergent for z ≤ 0, as is implied by (10). We obtain the needed result after taking the sth left derivative at z = 0.…”

Stern polynomials and double-limit continued fractions

“…Numerous properties of these polynomials can be found in [13]; here we only repeat the obvious properties (1.4) a(n; 0) = 1 (n ≥ 1), a(n; 1) = a(n) (n ≥ 0), and, for all m ≥ 0, (1.5) a(2 m ; x) = 1; this last identity follows by iterating (1.2). For ease of reference we also list the first Stern polynomials up to n = 32 in Table 1.…”

“…which is just Lemma 2.1 in [13]. Next we set again n = m + 1 in (2.2) and multiply both sides by x j , to obtain…”

“…Recently the authors [13] defined a polynomial analogue as follows. Let a(0; x) = 0, a(1; x) = 1, and for n ≥ 1 let a(2n; x) = a(n; x 2 ), (1.2) a(2n + 1; x) = xa(n; x 2 ) + a(n + 1; x 2 ).…”

“…Before we define these polynomials in Section 3, we prove a crucial result on the Stern polynomials which goes beyond [13] and is the basis of much of what follows in this paper; this will be done in Section 2. In Section 4 we derive several quadratic identities, and in Section 5 we define and investigate two analytic functions as limits of the above polynomial sequences.…”