Generalized polynomials and associated operational identities
Operational methods of Lie type in the theory of ordinary and generalized special functions, which are quasi-polynomials under the action of operators \(\widehat M= y-\widehat D^{-1}_x\), \(\widehat P= -\widehat D_x x\widehat D_x\), where \(D_x\) is the derivative and \(\widehat D^{-1}_x\) its inverse, offer a powerful tool to treat generating functions and the differential equations they satisfy and this is illustrated by considering the two variable Laquerre polynomial \[ L_n(x,y)= n! \sum^n_{k=0} {(-1)^k y^{n- k}x^k\over (n- k)!(k!)^2}. \] \(L_n(x,y)\) is shown to be a natural solution of \({\partial\over\partial y} L_n(x,y)= -\widehat D_xx \widehat D_x L_n(x,y)\) which is a kind of heat diffusion equation. In this paper, this method is extended to treat a new class of two variable Laguerre polynomials, called associated polynomials of second kind, their properties are discussed and the link between Laguerre type and Legendre polynomials is also discussed. A list of differential equations satisfied by the introduced polynomials is also given.
scientific article; zbMATH DE number 1367899