Polymeromorphic Itô–Hermite functions associated with a singular potential vector on the punctured complex plane

We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov–Bohm effect. The functions are defined by their β-modified Rodrigues type formula and extend the polyanalytic Itô–Hermite polynomials to the polymeromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.