International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 5, Pages 335-342
doi:10.1155/S0161171200001848

Mean number of real zeros of a random hyperbolic polynomial

Abstract

Consider the random hyperbolic polynomial, f(x)=1pa1coshx+⋯+np×ancoshnx, in which n and p are integers such that n≥2,   p≥0, and the coefficients ak(k=1,2,…,n) are independent, standard normally distributed random variables. If νnp is the mean number of real zeros of f(x), then we prove that νnp=π−1 logn+O{(logn)1/2}.