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

Abstract

From n= k=l Ft,n-l(ll,''', Ir)sgr-k, where 1 is a x matrix and 11,"', L are the invariants of (elementary symmetric functions of the eigenvalues), we first derive a formula for f(). Then we obtain the generating functions for the Fk,, and thence for the generalized Lucas polynomials F.,, n -l.

Functions of matrices

“…In several papers, some results related to (2.7) have been stated in terms of Bell and Lucas polynomials. See [4].…”

Operator Identities and the Solution of Linear Matrix Difference and Differential Equations

“…Taking some particular bases for !Jl!, we obtain the following formulas for the powers A k • then the funetions fj are solutions of (6.5) and their initial values are fj( k) = 0k,n-j'°~j~n, o~k -s n. ( 6.14) This result was proved by Brusehi and Ried [1], and it is related to the generalized Lueas polynomials.…”

A Study of the Second-Kind Multivariate Pseudo-Chebyshev Functions of Fractional Degree

“…It is well known (see, for example, [22,23]) that a basis for the r-dimensional vectorial space of solutions of the homogeneous linear bilateral recurrence relation with constant coefficients…”

“…It has been show by É Lucas [22,25] that all the {F k,n } n∈Z functions can be expressed through the bilateral sequence {F 1,n } n∈Z , corresponding to the initial conditions in Equation 4. More precisely, the following equations hold:…”