A distributional study of discrete classical orthogonal polynomials

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 (φΔPn+1 = anPn+2 + bnPn+1 + cnPn) also holds.