Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

A Theorem on Separated Transformations of Basis Vectors of Polynomial Space and Its Applications in Special Polynomials and Related Lie Algebra

Version 1 : Received: 25 February 2023 / Approved: 27 February 2023 / Online: 27 February 2023 (06:44:12 CET)

A peer-reviewed article of this Preprint also exists.

Manouchehr Amiri (2023). A Theorem on Separated Transformations of Basis Vectors of Polynomial Space and its Applications in Special Polynomials and Related sI(2, R) Lie Algebra. International Journal of Pure and Applied Mathematics Research, 3(1), 77-109. doi: 10.51483/IJPAMR.3.1.2023.77-109. Manouchehr Amiri (2023). A Theorem on Separated Transformations of Basis Vectors of Polynomial Space and its Applications in Special Polynomials and Related sI(2, R) Lie Algebra. International Journal of Pure and Applied Mathematics Research, 3(1), 77-109. doi: 10.51483/IJPAMR.3.1.2023.77-109.

Abstract

The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

Keywords

special polynomials; Hermite; Laguerre; Legendre; differential operators; Lie algebra; Baker-Campbell-Hausdorff formula; separated basis transformation; Forbenius covariant; Rodrigues formula; differential equations

Subject

Computer Science and Mathematics, Mathematics

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