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The bicomplex-real calculus and applications to bicomplex Hermite-Itô polynomials
Daniel Alpay1, Kamal Diki2, Mihaela Vajiac1
1Department of Mathematics, Chapman University, Orange, CA, USA.
Summary
This study introduces bicomplex Hermite polynomials and Landau operators, extending complex analysis to a new mathematical domain. These findings bridge classical complex theory with advanced bicomplex function theory.
Area of Science:
- Mathematics
- Complex Analysis
- Bicomplex Analysis
Background:
- Classical complex analysis and Hermite polynomials are foundational in mathematics.
- Extending these concepts to higher dimensions, such as bicomplex numbers, presents new theoretical challenges and opportunities.
Purpose of the Study:
- To extend complex C-R-calculus and complex Hermite polynomials to the bicomplex setting.
- To compare bicomplex polyanalytic function theory with the classical complex case.
- To introduce and analyze bicomplex Hermite polynomials and Landau operators.
Main Methods:
- Definition of bicomplex Hermite polynomials and exploration of their properties (Rodrigues formula, generating functions).
- Introduction of three bicomplex Landau operators.
- Calculation of the action of these operators on bicomplex Hermite polynomials.
Main Results:
- Successful extension of C-R-calculus and Hermite polynomials to the bicomplex domain.
- Characterization of two types of bicomplex Hermite polynomials with their fundamental properties.
- Demonstration of the application of bicomplex Landau operators on these polynomials.
Conclusions:
- The study establishes a foundation for bicomplex polyanalytic function theory.
- The introduced bicomplex analogues offer new tools for mathematical research.
- This work contributes to the broader themes of numerical analysis and orthogonal polynomials.
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