Appell Bases for Monogenic Functions of Three Variables

• Published in: Advances in applied Clifford algebras Vol. 23; no. 3; pp. 547 - 560
• Main Authors: Álvarez-Peña, Cynthia, Porter, R. Michael
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.09.2013
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Theoretical; Quaternionic analysis; Riesz system; Clifford analysis; Solid spherical harmonics; Monogenic function; Appell system; hyperholomorphic function; Complete orthonormal systems
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-013-0402-8

Monogenic (or hyperholomorphic) functions are well known in general Clifford algebras but have been little studied in the particular case R 3 → R 3 . We describe for this case the collection of all Appell systems: bases for the finite-dimensional spaces of monogenic homogeneous polynomials which respect the operator D = ∂ 0 − ∂ → . We prove that no purely algebraic recursive formula (in a specific sense) exists for these Appell systems, in contrast to the existence of known constructions for R 3 → R 4 and R 4 → R 4 . However, we give a simple recursive procedure for constructing Appell bases for R 3 → R 3 which uses the operation of integration of polynomials.