A Note on Bi-Orthogonal Polynomials and Functions

The theory of orthogonal polynomials is well established and detailed, covering a wide fieldofinterestingresults,as,inparticular,forsolvingcertaindifferentialequations. Ontheotherside the concepts and the related formalism of the theory of bi-orthogonal polynomials is less developed and much more limited. By starting from the orthogonality properties satisfied from the ordinary and generalized Hermite polynomials, it is possible to derive a further family (known in literature) of these kind of polynomials, which are bi-orthogonal with their adjoint. This aspect allows us to introduce functions recognized as bi-orthogonal and investigate generalizations of families of orthogonal polynomials.