Algebraic Polynomials with Random Coefficients with Binomial and Geometric Progressions

The expected number of real zeros of an algebraic polynomial ð ‘Ž ð ‘œ + ð ‘Ž 1 ð ‘¥ + ð ‘Ž 2 ð ‘¥ 2 + ⋯ + ð ‘Ž ð ‘› ð ‘¥ ð ‘› with random coefficient ð ‘Ž ð ‘— , ð ‘— = 0 , 1 , 2 , … , ð ‘› 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 ð ‘— th coefficient is v a r ( ð ‘Ž ð ‘—  ) = ð ‘› ð ‘—  . 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 ð ¸ ( ð ‘Ž ð ‘—  ) = ð ‘› ð ‘—  𠜇 ð ‘— + 1 and v a r ( ð ‘Ž ð ‘—  ) = ð ‘› ð ‘—  𠜎 2 ð ‘— . We show how the above expected number of real zeros is dependent on values of 𠜎 2 and 𠜇 in various cases.