Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 725260, 6 pages
Research Article

Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

1Department of Mathematics, University of Ulster at Jordanstown, County Antrim BT37 0QB, UK
2Department 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 polynomial ao+a1x+a2x2++anxn with random coefficient aj,j=0,1,2,,n is known. The distribution of the coefficients is often assumed to be identical albeit allowed to have different classes of distributions. For the nonidentical case, there has been much interest where the variance of the jth coefficient is var(aj)=(nj). It is shown that this class of polynomials has significantly more zeros than the classical algebraic polynomials with identical coefficients. However, in the case of nonidentically distributed coefficients it is analytically necessary to assume that the means of coefficients are zero. In this work we study a case when the moments of the coefficients have both binomial and geometric progression elements. That is we assume E(aj)=(nj)μj+1 and var(aj)=(nj)σ2j. We show how the above expected number of real zeros is dependent on values of σ2 and μ in various cases.