In this talk, some aspects of the theory of orthogonal polynomials will be in discussion. The elements of a classical polynomial sequence (Hermite, Laguerre, Bessel and Jacobi) are eigenfunctions of a second order linear differential operator with polynomial coefficients L, known as the Bochner’s operator. In an algebraic manner, a classical sequence is also characterised through the so-called Hahn’s property, which states that an orthogonal polynomial sequence is classical if and only if the sequence of its (normalised) derivatives is also orthogonal.

To begin with, it is shown that an orthogonal polynomial sequence (OPS) is classical if and only if any of its polynomials fulfils a certain differential equation of order 2k, for some positive integer k. The structure of such differential equation is thoroughly revealed, permitting to explicitly present the corresponding 2k-order differential operator L_k. On the other hand, as a consequence of Bochner’s result, any element of a classical sequence must be an eigenfunction of a polynomial with constant coefficients in powers of L. With the introduction of the so-called A-modified Stirling numbers (where A indicates a complex parameter), we are able to establish inverse relations between the powers of the Bochner operator L and L_k .

The second part of this talk is focused on a generalization on the Hahn’s problem. Given certain lowering operators O (linear operators that decrease in one unit the degree of a polynomial), we will expound the search of all the O-classical sequences, in other words, all the orthogonal polynomial sequences {P_n}_n O such that {OP_n}_n≧0 is also orthogonal.