The Matrix Operator e_x and the Lucas Polynomials

Iterative Formulas Associated with Generalized Third Order Recurrence Relations

“…It follows by use of the power series expansion for the right-hand side of (4.6) in When R is zero, the above results agree with those of Barakat [1] and Williams [6]. Other approaches which do not depend on the explicit solution of the cubic auxiliary equation can be based on the use of the generalized second order sequence of numbers, {w,}, discussed by Horadam [4], and either use Cardan's solution of the cubic to reduce it to the quadratic auxiliary equation of {wn} as in Williams [6], or use the shift operator E to reduce the third order numbers to second order numbers as in Shannon and Horadam [5].…”

An Explicit Formula for $f(\mathcal{A})$ and the Generating Functions of the Generalized Lucas Polynomials

“…Note that (ii) implies Sylvester's matrix interpolation formula and does not in general require the knowledge of the Jordan canonical form of d. Recently formulae for e only have been given [1], [16], [12].…”

“…., r, and therefore of the generalized Lucas polynomials (in r variables) k-1 E Fg,n+r-g-1 (I1,''' Ir)Z n= Various papers have been devoted to the study of the above polynomials (see [21]) and to the extension of the algebraic theory of the Lucas numerical functions (see, e.g., [133, [22]). 1 The functions Fl.n(I1,'", L), n--1, are called in the literature generalized Lucas polynomials (see [2], [21]). The above results have been extended (see [4]) to a matrix whose minimal polynomial is known.…”