ON SEMICLASSICAL LINEAR FUNCTIONALS - INTEGRAL-REPRESENTATIONS

If L is a functional of moments and satisfies the equation D(phi L)+psi L = 0, where phi(x) and psi(x) are arbitrary polynomials but with the only condition deg(psi) greater than or equal to 1, then L is said to be a semiclassical functional. When L is semiclassical and regular, its corresponding orthogonal polynomial sequence (OPS), (P-n(x))(n=0)(infinity), is called a semiclassical OPS. In this paper integral representations are given for semiclassical functionals, regular or not, in cases: (A) deg(phi) > deg(psi) and (B) deg(phi) less than or equal to deg(psi), but in the latter case when deg(phi)= 0. The fundamental result is the following: if mu(n), with n greater than or equal to 0, are the moments of an (A)-functional then \mu(n)\ less than or equal to CM(n). This allows us to find integral representations for (A)-functionals using standard techniques taken from Pollaczek (1951).