Generalized Humbert polynomials and second-order partial differential operators

We define a new class of Humbert's polynomials which generalizes the well-known class of Gegenbauer, Legendre, Pincherle, Horadam–Mahon, Kinney, Hordam–Pethe, Gould and Pathan–Khan polynomials. We prove that these generalized polynomials are eigenfunctions of the Hamiltonian of the Calogero–Sutherland model. We show several differential equations having these polynomials for solutions.