Year: 2026
Type: preprint
Abstract: A new class of algebras with variable operator-valued structural constants is introduced. In these algebras, the structural operators are allowed to depend on the coordinates of the hypercomplex variable as well as on additional parameters. A theory of monogenic functions is developed for a two-dimensional commutative associative operator algebra with variable structural operators. Generalized Cauchy–Riemann conditions are obtained, and a compatibility system for the structural operators is established. It is shown that, under the compatibility conditions, monogenic functions are closed with respect to addition and multiplication, which provides a method for constructing new monogenic functions, including hypercomplex polynomials. A relationship between monogenic functions and second-order operator-differential equations with variable coefficients is established. It is proved that the components of monogenic functions satisfy certain operator-differential equations, and converse results are obtained, providing a constructive correspondence between solutions of these equations and monogenic functions. By realizing the structural operators on appropriate function spaces, the proposed approach makes it possible to pass from operator-differential equations to various classes of differential, integro-differential, and functional-differential equations for scalar- and vector-valued functions. Several examples illustrating this construction are presented.
Source: arXiv (Cornell University)
Author: Shpakivskyi, Vitalii
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Science Foundation Ireland
OpenAlex ID: W7211907537