Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial+1+22+⋯+ with random coefficient,=0,1,2,…, 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 th coefficientis var()=. 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 ()=+1 and var()=2. We show how the above expected number of real zeros isdependent on values of 2 and in various cases