Polynomial sequences generated by integral powers of differential operators

<p>Polynomial sequences generated by integral powers of first and second order differential operators conveniently chosen will be the issue. More precisely, the focus will lie on their connection with well known orthogonal polynomial sequences along with their foremost structural properties. This talk will be split in two parts. We will start by analysing the cases in which the aforementioned differential operator is of first order, bringing into analysis polynomial sequences associated to the classical linear functionals of Hermite, Laguerre, Bessel and Jacobi. Afterwards, the discussion will proceed towards the analysis of polynomial sequences generated by second order differential operators, which brings up the open problem of characterizing orthogonal polynomial sequences with respect to certain positive definite linear functionals. The Kontorovich-Lebedev transform and the central factorial numbers will be an asset to attain our goals.</p> <h4>References</h4> <ol> <li>Ana F. Loureiro, New results on the Bochner condition about classical orthogonal polynomials, J. Math An. Appl., 364 (2010) 307-323.</li> <li>Ana F. Loureiro, P. Maroni, S. Yakubovich, On a nonorthogonal polynomial sequence associated with Bessel operator, Pre-Print CMUP 2011-10 (ArXiv:1104.4055v1)</li> <li>Ana F. Loureiro, S. Yakubovich, On a polynomial sequence related to the Ditkin-Prudnikov problem, Pre-Print CMUP 2011-23 (arXiv:1110.6015v1)</li> </ol>