Explicit formulas for the distribution of complex zeros of a family of random sums

The present paper provides an explicit formula for the average intensity of the distribution of complex zeros of a family of random sums of the form Sn(z)=∑j=0nηjfj(z), where z   is the complex variable x+iyx+iy, ηj=αj+iβjηj=αj+iβj and {αj}j=0n and {βj}j=0n are sequences of standard normal independent random variables, and {fj}j=0n is a sequence of given analytic functions that are real-valued on the real number line. In addition, the numerical computations of the intensity functions and the empirical distributions for the special cases of random Weyl polynomials, random Taylor polynomials and random truncated Fourier cosine series are included as examples.