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.)
The linear functional u is said to be regular if there exists a
monic polynomial sequence [{B.sub.n]}.sub.n[greater than or equal to]0] such that
Since V(0) [not equal to] 0, and since the roots of a
monic polynomial identify it uniquely, the factorization in (1) is unique.
where E and F are non-zero polynomials, with E a
monic polynomial.
This implies that we can as well consider the
monic polynomialLet [q.sub.n] (x) be the classical
monic polynomial, orthogonal with respect to a classical weight function w (x) on (a, b).
