A note on the arithmetic properties of Stern Polynomials

We investigate the Stern polynomials defined by $B_0 ( t ) =0,B_1 ( t ) =1$, and for $n \geq 2$ by the recurrence relations $B_{2n}( t) =tB_{n}( t) ,$ $B_{2n+1}( t) =B_n( t) +B_{n+1}( t)$. We prove that all possible rational roots of that polynomials are $0,-1,-1/2,-1/3$. We give complete characterization of $n$ such that $deg( B_n) = deg( B_{n+1})$ and $deg( B_n) =deg( B_{n+1}) =deg( B_{n+2})$. Moreover, we present some result concerning reciprocal Stern polynomials.Comment: 9 pages, submitte

Zeros and convergent subsequences of Stern polynomials

We investigate Dilcher and Stolarsky's polynomial analogue of the Stern diatomic sequence. Basic information is obtained concerning the distribution of their zeros in the plane. Also, uncountably many subsequences are found which each converge to a unique analytic function on the open unit disk. We thus generalize a result of Dilcher and Stolarsky from their second paper on the subject.Comment: 10 pages, 1 figur

On certain arithmetic properties of Stern polynomials

We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. We study also the sequence e(n)=\op{deg}_{t}B_{n}(t) and give various of its properties.Comment: 20 page

Arithmetic properties of the sequence of degrees of Stern polynomials and related results

Let $B_{n}(t)$ be a $n$-th Stern polynomial and let e(n)=\op{deg}B_{n}(t) be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\{e(n)\}_{n=1}^{\infty}$. We also study the sequence d(n)=\op{ord}_{t=0}B_{n}(t). Among other things we prove that $d(n)=\nu(n)$, where $\nu(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. We also obtain an interesting closed expression for a certain sum involving Stern polynomials.Comment: 16 page

The Stern diatomic sequence via generalized Chebyshev polynomials

Let a(n) be the Stern's diatomic sequence, and let x1,...,xr be the distances between successive 1's in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1+1, ..., xr+1, and we derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity. We also show that a(n) = Det(Ir + Mr), where Ir is the rxr identity matrix, and Mr is the rxr matrix that has x1,...,xr along the main diagonal, then all 1's just above and below the main diagonal, and all the other entries are 0