A distributional study of discrete classical orthogonal polynomials

Abstract

For the sequences of discrete classical orthogonal polynomials (Charlier, Meixner, Hahn) we can find a functional u, which satisfies the difference distributional equation Δ( φu) = ψu where φ and ψ are polynomials of degrees ⩽2 and 1 respectively. From this it follows that these polynomials are solutions of a second-order difference equation; also, they can be represented by a Rodrigues-type formula. The sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence. Their weight functions satisfy a Pearson-type difference equation. A structure relation ( φΔP n+1 = a n P n+2 + b n P n+1 + c n P n ) also holds.