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Search: MSC category 11B83 ( Special sequences and polynomials )

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1. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
On the Convergence of a Class of Nearly Alternating Series
Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $ \alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$.

Keywords:Series, convergence, almost alternating, convex, continued fractions
Categories:40A05, 11A55, 11B83

2. CJM 2001 (vol 53 pp. 33)

Borwein, Peter; Choi, Kwok-Kwong Stephen
Merit Factors of Polynomials Formed by Jacobi Symbols
We give explicit formulas for the $L_4$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials $$ f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n. $$ and $$ f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n. $$ where $(\frac{\cdot}{N})$ is the Jacobi symbol. Two cases of particular interest are when $N = pq$ is a product of two primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt, Jensen and Jensen and of the authors. This study arises from a number of conjectures of Erd\H{o}s, Littlewood and others that concern the norms of polynomials with $-1,1$ coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite~$N$.

Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols
Categories:11J54, 11B83, 12-04

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