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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.

It is well known that an analytic function f of a matrix

is assumed for defining (and computing)

Let

the polynomial interpolating

If the eigenvalues are all distinct,

This avoids the use of higher powers of

In the following, we show how to proceed in connections with the matrix

We propose an alternative method, based on recursion, using the functions

Another Taylor series will be used, but using only functions of the invariants of

It is worth to recall that the knowledge of eigenvalues is equivalent to that of invariants, since the latter are the elementary symmetric functions of the former (with alternate sign).

Up to our knowledge, this is the first time that polynomials are used to solve this kind of differential problems, furthermore our method has the advantage to avoid computation of higher powers of the matrix

It is well-known that a higher order differential system can be always be reduced to a first order system, hence we will limit ourselves to considering such type of systems.

For simplicity, we start off with the

with matrix

assuming

In the following, consider the Cauchy problem with initial conditions:

Looking at the first equation in (2.1), we note that since the right hand side is (real or) complex analytic, the solution is (real or) complex analytic as well. Deriving side by side, we find

Eliminating

This procedure can be iterated, obtaining, for example

In general we find the same recursion satisfied by the powers of the matrix

where the coefficients

and the initial conditions:

It is easily shown that the second function

As a consequence, putting

and using Taylor expansion, the solution of the Cauchy problem (2.1) - (2.2), can be found in the form:

The above result can be put in vectorial form, in order to be generalized.

Let

Introduce the matrix

then, the solution in vectorial form reads

Note that the convergence of the vectorial series in any compact set K of the space

In the following section, we will extend this solution to the general vectorial case.

Remark 2.1 Note that Equation (9) does not use all powers of matrix

Now, we consider also the case of the

with matrix

we suppose

We consider, the Cauchy problem with initial conditions:

By using the same technique as in the

and by iterating the procedure we obtain, for example,

and so on. In general we find the same recursion satisfied by the powers of the matrix

where the coefficients

and the initial conditions:

The second and third function

As a consequence, putting

and

and using Taylor expansion, the solution of the Cauchy problem (3.1) - (3.2), can be found in the form:

The above result can be put in the following vectorial form.

Let

Introduce the matrix

then, the solution in vectorial form reads

Remark 3.1 Even in this case, the considerations of Rem. 2.1 still hold, showing a more convenient form of computing solutions of the Cauchy problem (3.1) - (3.2), with respect to traditional methods, as reported e.g. in [

Theorem 4.1 Consider the Cauchy problem for a homogeneous linear differential system

where

denote by

Suppose that the system cannot be reduced to a lower order system, so that

Introduce the matrix

then, the solution of problem (3.1) takes the form

Proof―The proof can be found by induction, considering the r vector

composed of its first component

Note that the convergence of the vectorial series in any compact set K of the space

We have recalled that the exponential

Furthermore, by using the functions

Therefore, this is, in our opinion, a more convenient technique for solving problem (4.1).

PierpaoloNatalini,Paolo E.Ricci, (2015) A “Hard to Die” Series Expansion and Lucas Polynomials of the Second Kind. Applied Mathematics,06,1235-1240. doi: 10.4236/am.2015.68116