The bicomplex-real calculus and applications to bicomplex Hermite-Itô polynomials.

Abstract

In this paper, we extend the notions of complex C-R-calculus and complex Hermite polynomials to the bicomplex setting and compare the bicomplex polyanalytic function theory with the classical complex case. Specifically, we introduce two kinds of bicomplex Hermite polynomials and present some of their basic properties, such as the Rodrigues formula and generating functions. We also define three bicomplex Landau operators and calculate their action on the bicomplex Hermite polynomials of the first kind.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.