Real zeros of a random sum of orthogonal polynomials

Abstract:

Let ${c_0},{c_1},{c_2}, \cdots$ be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let $P_k^ \ast (x)(k = 0,1,2, \cdots )$ be the normalised Legendre polynomials orthogonal with respect to the interval $( - 1,1)$. It is proved that the average number of the zeros of ${c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x)$ in the same interval is asymptotically equal to ${(3)^{ - 1/2}}n$ when $n$ is large.