REAL ZEROS OF A RANDOM POLYNOMIAL WITH LEGENDRE ELEMENTS

Let T;(x),T(x),...,T(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (-1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial goT(x) + glT(x) +... + gnT(x) where gj, j 0,1,...,n are independent identically and normally distributed random variables with mean zero and variance one is known. The present paper considers the case when the means and variances of the coefficients are not all necessarily equal. It is shown that in general this expected number of real zeros is only dependent on variances and is independent of the means.


Introduction
Let {gj(w)}= o be a sequence of independent normally distributed random variables defined on a probability space (fl, J,Pr).Let Nn(a b) be the number of real zeros of Pn(x) in the interval (a,b)where in which Pn(x) Pn(x'w) E gj(w) T(x)   .=0 (1.1) T(x) v/j + 1/2)Tj(x), and Tj(x) is a Legendre polynomial and therefore T(x)is a normalized Legendre polynomial orthogonal with respect to the weight function unity.For the case of identical normal standard distributed coefficients Das [3] shows that ENn(-1,1) n/V/when n is sufficiently large.Recently in an interesting paper using a delicate method, Wilkins [12] shows that ENn(-1,1)-n/v/+o(n5) for any positive 5.
Here we consider the case when the means and variances of the coefficients of 258 K. FARAHMAND (1.1) are not all equal.We show that ENn(-1, 1)is independent of E(gj) but depen- dent on var(gj).This is in complete contrast to the result obtained by Farahmand   [5] for the random algebraic polynomial ,=ogj(w)xJ, who showed that ENn(-c,oc) is independent of variances and dependent on the means.Neverthe- less, as far as EN n is concerned there are similarities between polynomials of type (1.1) and random trigonometric polynomials =ogj(w)cosjx, see for example [4], in that both have O(n) number of real zeros and both are effected by the variance of the coefficients and not by their means.A survey of the earlier works together with comprehensive references on the subject can be found in Bharucha-Reid and Sambandham [1].
The result in Theorem 3 does, indeed, correspond to that of Das [3], and therefore it shows that Das' result remains valid for polynomials with non-identical distributed coefficients.These non-identical cases, of interest in their own right, are important as they lead to the expected number of crossings of two polynomials with different de- grees.Let n n Fn(x fj(w)T(x) and Qn(X)qj(w)T(x).j=0 3=0 Then, for n > n', the expected number of real zeros of polynomial Fn(x Qn,(X) {fj(w) qj(co)}T(x) + fj(w)T(x), j=0 j=n can serve as the expected number of crossings of Fn(x by Qn,(X).On the other hand, Fn(x)-Qn,(x can be represented as n_._ogj(w)T(xwhere gj(w)= fj(w)-qj(w) for 0 < j _< n' and gj(co) qj(w) for nZ j _< n, which is, indeed, in the form of Pn(x) studied in Theorem 1-3.
Then from the Darboux-Christoffel formula [6, (k + 1)(2k + 3)1/2/2(2k + 1) (2.7) At this stage we abandoned the calculations of A1 and A2 as only an upper limit for these are required.We proceed to evaluate these upper limits as well as the domin- ant terms for A2, B 2 and C in the following section.

Proofs of Theorems
We first use the Kac-Rice formula (2.4) for ENn(-l+ ,1-).We will see that this interval yields the main contribution to the expected number of real zeros.The expected number of real zeros outside this interval, which, it so happens, are negligi- ble, is estimated by using an application of :ensen's theorem. as