Average Number of Real Roots of Random Harmonic Equations

Let {g_k}be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let _k(t ) (k = 0, 1, 2,...) be the normalized Jacobi polynomials orthogonal with respect to the interval [ – 1, 1 ]. Then it is proved that the average number of real roots of the random equations,_k=0ⁿg_kk(1)=C where Cis a constant, is asymptotically equal to n/in the same interval when nis large and even for C as long as C=O (n²).