On semiclassical linear functionals: integral representations

If is a functional of moments and satisfies the equation D (φ)+ψ=o, where φ(x) and ψ(x) are arbitrary polynomials but with the only condition deg(ψ) ⩾ 1, then is said to be a semiclassical functional. When is semiclassical and regular, its corresponding orthogonal polynomial sequence (OPS), (Pn(x)n=0∞ is called a semiclassical OPS. s paper integral representations are given for semiclassical functionals, regular or not, in cases: (A) deg(φ) > deg(ψ) and (B) deg(φ) ⩽ deg(ψ), but in the latter case when deg(φ) = 0. ndamental result is the following: if μn, with n ⩾ 0. are the moments of an (A)-functional then ¦μn¦ ⩽ CMn. This allows us to find integral representations for (A)-functionals using standard techniques taken from Pollaczek (1951).