Since [c.sub.1](z), [c.sub.2](z), ..., [c.sub.m](z) are arbitrary monic polynomials
subject, however, to conditions (6) and (7), we assume without any loss of generality that
(1) We are dealing only with monic polynomials
, that is, with those for which the coefficient of [x.sup.2] is 1.
By using (1.1) we obtain a system of |n| linear equations for the |n| unknown coefficients of the monic polynomial
[P.sub.n](x) = [[summation].sup.|n|.sub.k=0] [a.sub.k],nxk, [a.sub.|n|,n] = 1.
Our goal here is the first interpretation for the coefficients of [[PHI].sub.n](x), which applies more generally to the coefficients of any monic polynomial
f(x) in Z[x].
Such a monic polynomial
may not exist or may not be unique.
A sequence of monic polynomials
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is said to be a Laguerre-Hahn sequence of class s if it is orthogonal with respect to a linear functional w satisfying
(To justify this claim, the main things to note are that (a) a monic polynomial
that is irreducible over Z is also irreducible over [Z.sub.2] = [F.sub.2], and (b) if P' is P with coefficients reduced mod 2, and H(P',Q') [less than or equal to] d, then by adding even integers to the coefficients of Q' we can arrive at a polynomial Q such that |P - Q| [less than or equal to] d.)
Since V(0) [not equal to] 0, and since the roots of a monic polynomial
identify it uniquely, the factorization in (1) is unique.
This implies that we can as well consider the monic polynomial
Let [q.sub.n] (x) be the classical monic polynomial
, orthogonal with respect to a classical weight function w (x) on (a, b).