Abstract

The present paper introduces a method of basis transformation of a vector space that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method is based on separated transformations 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 Sí (2, R), 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