Solution of the Difference Equations of Generalized Lucas Polynomials

Abstract

Barakat and Baumann have introduced polynomials U(N) (a1, a2, …, aN) termed the generalized Lucas polynomials satisfying a difference equation with a set of initial conditions. We show that these polynomials can be obtained directly from the symmetric functions hn, which are of basic importance in combinatorial analysis. Moreover, we extend the definition of V(a1, a2) to V(N) (a1, a2, …, aN) and establish that these polynomials too can be obtained from the symmetric functions Sn. Further, closed expressions for the U and V are obtained.