Theorems on Separated Transformations of Basis Vectors of Polynomial Space and its Applications in Classical Orthogonal Polynomials

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. This method is based on separated transformations of vector space basis by a set of operators that their overall action is equivalent to the formal basis transformation and connected to it by linear combination with projection operators. By introducing the new relations for projection operators under separated transformation and applying the Forbenius covariants as projection operators, we derive a general method that incorporates the Rodrigues formula as a special case in polynomial space. A solution method to some classical differential equation such as Hermite and Laguerre differential equations is presented.