Expected Number of Real Zeros for Random Linear Combinations of Orthogonal Polynomials

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1)) logn expected real zeros in terms of the degree n. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have n/ √ 3 + o(n) expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros. 1. Background Zeros of polynomials with random coefficients have been intensively studied since 1930s. The early work concentrated on the expected number of real zeros E[Nn(R)] for polynomials of the form Pn(x) = ∑n k=0 ckx , where {ck}k=0 are independent and identically distributed random variables. Apparently the first paper that initiated the study is due to Bloch and Pólya [2]. They gave an upper bound E[Nn(R)] = O( √ n) for polynomials with coefficients selected from the set {−1, 0, 1} with equal probabilities. Further results generalizing and improving that estimate were obtained by Littlewood and Offord [21]-[22], Erdős and Offord [8] and others. Kac [17] established the important asymptotic result E[Nn(R)] = (2/π + o(1)) log n as n→∞, for polynomials with independent real Gaussian coefficients. More precise forms of this asymptotic were obtained by many authors, including Kac [18], Wang [32], Edelman and Kostlan [7]. It appears that the sharpest known version is given by the asymptotic series of Wilkins [33]. Many additional references and further directions of work on the expected number of real zeros may be found in the books of Bharucha-Reid and Sambandham [1], and of Farahmand [9]. In fact, Kac [17]-[18] found the exact formula for E[Nn(R)] in the case of standard real Gaussian coefficients: E[Nn(R)] = 4 π ∫ 1 0 √ A(x)C(x)−B2(x) A(x) dx,