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
Department of Mathematics, Clark Atlanta University, Atlanta 30314, GA, USA
Received 25 March 1998
Copyright © 2000 J. Ernest Wilkins. 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.
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}.