New results on the Bochner condition about classical orthogonal polynomials (Q847700)

The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences whose elements are eigenfunctions of the second-order Bochner differential operator \(\mathcal F\). In [\textit{A. F. Loureiro, P. Maroni} and \textit{Z. Da Rocha}, J. Math. Anal. Appl. 322, No.~2, 645--667 (2006; Zbl 1106.33009)] these polynomials were described as eigenfunctions of an even order differential operator \(\mathcal F_k\) with polynomial coefficients defined by a recursive relation. Here, an explicit expression of \(\mathcal F_k\) for any positive integer \(k\) is given. The main aim of this work is to establish explicitly sums relating any power of \(\mathcal F\) with \(\mathcal F_k\), \(k\geq 1\). In other words, to expose a pair of inverse relations between these two operators.