Given a polynomial f (x) = a(0)x(n) + a(1)x(n-1) + ... + a(n) with positive coefficients a(k), and a positive integer M <= n, we define an infinite generalized Hurwitz matrix H-M(f) : = (a(Mj-i))(i, j). We prove th...
详细信息
Given a polynomial f (x) = a(0)x(n) + a(1)x(n-1) + ... + a(n) with positive coefficients a(k), and a positive integer M <= n, we define an infinite generalized Hurwitz matrix H-M(f) : = (a(Mj-i))(i, j). We prove that the polynomial f (z) does not vanish in the sector {z is an element of C : vertical bar arg(z)vertical bar < pi/M} whenever the matrix H-M is totally non-negative. This result generalizes the classical Hurwitz' Theorem on stable polynomials (M = 2), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots (M = 1), and the Cowling-Thron theorem (M = n). In this connection, we also develop a generalization of the classical euclideanalgorithm, of independent interest per se.
We exhibit and study various regularity properties of the sequence (R (n))(n greater than or equal to 1) which counts the number of different representations of the positive integer n in the Fibonacci numeration syste...
详细信息
We exhibit and study various regularity properties of the sequence (R (n))(n greater than or equal to 1) which counts the number of different representations of the positive integer n in the Fibonacci numeration system. The regularity properties in question are observed by representing the sequence as a two-dimensional array consisting of an infinite number of rows L-1, L-2, L-3.... where each L-k contains f(k-1) (the k - 1st Fibonacci number) entries of the sequence (R (n)). We give a purely combinatorial recursive algorithm for generating each row L-k from previous rows L-j with j < k. We then show that for each positive integer m, and for all k greater than or equal to 2m, the number of occurrences of m in L-k is a constant rk(m) depending only on m. The function rk(m) has many interesting number theoretic properties and is intimately connected to the Euler phi-function. (C) 2004 Elsevier B.V. All rights reserved.
暂无评论