Connection relations and characterizations of orthogonal polynomials

Abstract

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam–Chihara, Askey–Wilson, and Meixner–Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obermaier. Along the way we explicitly evaluate the connection coefficients in the expansion of Dq2pn in terms of {pk}, where Dq is the Askey–Wilson operator and {pk} are general Askey–Wilson polynomials. As a limiting case we derive the corresponding connection coefficients in the expansion of W2Wn in terms of {Wk}, where W is the Wilson operator and {Wk} are general Wilson polynomials. Using the connection relation for Askey–Wilson polynomials, we obtain a characterization for the two-parameter symmetric Askey–Wilson polynomials. The connection relations between DmPn(α,β), D:=d/dx and {Pk(α,β)} are also derived.