Journal of Applied Mathematics and Stochastic Analysis
Volume 2007 (2007), Article ID 43091, 7 pages
doi:10.1155/2007/43091
Research Article
On Zeros of Self-Reciprocal Random Algebraic Polynomials
Department of Mathematics, University of Ulster, Jordanstown, Co. Antrim BT37 0QB, UK
Received 21 June 2007; Accepted 31 October 2007
Copyright © 2007 K. Farahmand. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper provides an asymptotic estimate for the expected number
of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random
algebraic polynomials with complex variables and complex random coefficients,
with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.