Applied Mathematics, 2015, 6, 1235-1240 Published Online July 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.68116 How to cite this paper: Natalini, P. and Ricci, P.E. (2015) A “Hard to Die” Series Expansion and Lucas Polynomials of the Second Kind. Applied Mathematics, 6, 1235-1240. http://dx.doi.org/10.4236/am.2015.68116 A “Hard to Die” Series Expansion and Lucas Polynomials of the Second Kind Pierpaolo Natalini 1* , Paolo E. Ricci 2 1 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Roma, Italia 2 International Telematic University UNINETTUNO, Roma, Italia Email: * natalini@mat.uniroma3.it , paoloemilioricci@gmail.com Received 5 June 2015; accepted 10 July 2015; published 14 July 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix. Keywords Homogeneous Linear Differential Systems with Constant Coefficients, Exponential Matrix, Lucas Polynomials of the Second Kind 1. Introduction It is well known that an analytic function f of a matrix r r × =   , i.e. ( ) f  is the matrix polynomial obtained from the scalar polynomial interpolating the function f on the eigenvalues of  (see e.g. the Gantmacher book [1]), however, in many books (see e.g. [2]), the series expansion 0 exp ! n n n ∞ = = ∑   (1.1) is assumed for defining (and computing) ( ) exp  . So, apparently, the series expansion for the exponential of a matrix is “hard to die”. Let { } 1 2 , , , r λλ λ Σ=  be the spectrum of  . Denoting by ( ) 1 1 0 1 1 r r r P x x a x α α − − − = + + +  the polynomial interpolating ( ) exp x on Σ , i.e. such that: ( ) ( ) 1 exp n i i P λ λ − = , ( ) for 1, 2, , i r =  , then * Corresponding author.