Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

[摘要] The expected number of real zeros of an algebraic polynomialao+a1x+a2x2+⋯+anxnwith random coefficientaj,j=0,1,2,…,nis 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 thejth coefficientisvar  (aj)=(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 assumeE(aj)=(nj)μj+1andvar  (aj)=(nj)σ2j. We show how the above expected number of real zeros isdependent on values ofσ2andμin various cases.

[学科分类] 应用数学