Generalized Gould-Hopper Polynomials by Emil Horozov

Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems with this property. In a recent paper [22] we extended the class of polynomial systems having both the property to be eigenfunctions of a differential operator (now without restriction on the order) and the 3-term recurrence relation, responsible for the orthogonality is relaxed to a d+2-term relation with \(d \in \mathbb{N}\). We found such systems also when instead of a differential operator we consider a difference one. Our polynomial systems came with many properties. In the present paper we continue to study a class of these systems in the spirit of the classical orthogonal polynomials. Their most important properties, are hypergeometric representations that we find for them. From the hypergeometric representations we derive generating functions and in some cases we find Mehler-Heine type formulas.

The paper is available here.

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