Algebraic Polynomials with Random Coefficients with Binomial and Geometric ProgressionsReportar como inadecuado

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Journal of Applied Mathematics and Stochastic AnalysisVolume 2009 2009, Article ID 725260, 6 pages

Research Article

Department of Mathematics, University of Ulster at Jordanstown, County Antrim BT37 0QB, UK

Department of Mathematics, Morehouse College, Atlanta, GA 30114, USA

Received 28 January 2009; Accepted 26 February 2009

Academic Editor: Lev Abolnikov

Copyright © 2009 K. Farahmand and M. Sambandham. 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.


The expected number of real zeros of an algebraic polynomial𝑎𝑜+𝑎1𝑥+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛 with random coefficient𝑎𝑗,𝑗=0,1,2,…,𝑛 is known. The distribution of thecoefficients is often assumed to be identical albeit allowed tohave different classes of distributions. For the nonidentical case,there has been much interest where the variance of the 𝑗th coefficientis var𝑎𝑗=𝑛𝑗. It is shown that this classof polynomials has significantly more zeros than the classicalalgebraic polynomials with identical coefficients. However,in the case of nonidentically distributed coefficients it isanalytically necessary to assume that the meansof coefficients are zero. In this work westudy a case when the moments of the coefficients have bothbinomial and geometric progression elements. That is we assume 𝐸𝑎𝑗=𝑛𝑗𝜇𝑗+1 and var𝑎𝑗=𝑛𝑗𝜎2𝑗. We show how the above expected number of real zeros isdependent on values of 𝜎2 and 𝜇 in various cases.

Autor: K. Farahmand and M. Sambandham



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