Characterization and construction of classical orthogonal polynomials using a matrix approach

We introduce a new algebraic approach for the study of orthogonal polynomial sequences. We use simple properties of infinite matrices to obtain some basic results in the theory of orthogonal polynomial sequences, including several characterizations of the classical sequences, some of which are equivalent to the ones obtained by Bochner, Maroni, and Al Salam and Chihara. We also obtain explicit formulas for the coefficients of the recurrence relations of all the classical orthogonal polynomials in terms of four parameters. The polynomial coefficients and the eigenvalues that appear in Bochner’s differential equation are also found explicitly.