New results on the Bochner condition about classical orthogonal polynomials

The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences (OPS) whose elements are eigenfunctions of the Bochner second-order differential operator F (Bochner, 1929 [3]). In Loureiro, Maroni and da Rocha (2006) [18] these polynomials were described as eigenfunctions of an even order differential operator Fk with polynomial coefficients defined by a recursive relation. Here, an explicit expression of Fk for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of F with Fk, k?1, in other words, to bring a pair of inverse relations between these two operators. This goal is accomplished with the introduction of a new sequence of numbers: the so-called A-modified Stirling numbers, which could be also called as Bessel or Jacobi–Stirling numbers, depending on the context and the values of the complex parameter A.