General Linear Recurrence Sequences and Their Convolution Formulas

: We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to ﬁnd convolution formulas for second order linear recurrence polynomials generated by (cid:16) The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.


Introduction
Generating functions [1] constitute a bridge between continuous analysis and discrete mathematics. Linear recurrence relations are satisfied by many special polynomials of classical analysis. A wide scenario including special sequences of polynomials and numbers, combinatorial analysis, and application of mathematics is related to the above mentioned topics.
It would be impossible to list in the Reference section all of even the most important articles dedicated to these subjects. As a first example, we recall the Chebyshev polynomials of the first and second kind, which are powerful tools used in both theoretical and applied mathematics. Their links with the Lucas and Fibonacci polynomials have been studied and many properties have been derived. Connections with Bernoulli polynomials have been highlighted in [2].
In particular, the important calculation of sums of several types of polynomials have been recently studied (see e.g., [3][4][5] and the references therein). This kind of subject has attracted many scholars. For example, W. Zhang [6] proved an identity involving Chebyshev polynomials and their derivatives.
Fibonacci and Lucas polynomias and their extensions have been studied for a long time, in particular within the Fibonacci Association, which has contributed to the study of this and similar subjects. As an applications of a results proved by Y. Zhang and Z. Chen [3], Y. Ma and W. Zhang [4] obtained some identities involving Fibonacci numbers and Lucas numbers.
Convolution techniques are connected with combinatorial identities, and many results have been obtained in this direction [2,7,8]. Convolution sums using second kind Chebyshev polynomials are contained in [7].
for x ∈ R and r ∈ N, they proved the interesting relation Furthermore, they derived a link between p n (x) and a particular combination of sums of Fibonacci numbers, so that complex sums of Fibonacci numbers have been converted to the easier calculation of p n (x).
In a recent article Chen Zhuoyu and Qi Lan [9] introduced convolution formulas for second order linear recurrence sequences related to the generating function [1] of the type deriving coefficient expressions for the series expansion of the function f x (t), (x ∈ R). In this article, motivated by this research, we continue the study of possible applications of the considered method, by analyzing the general situation of a generating function of the type G(t, x) = 1 1 + a 1 t + a 2 t 2 + · · · + a r t r x , and we deduce the recurrence relation for the generated polynomials.
Several illustrative examples are shown in Section 6. In the last section the results are extended, in a straightforward way, to the case of matrix polynomials.

Generating Functions
We start from the generating function considered by Chen Zhuoyu and Qi Lan: with where Note that, by Equation (2) we could write, in equivalent form: but, in what follows, we put for shortness: By Equations (3), (4a) and (4b) we find the convolution formula:

Recurrence Relation
Note that as can be derived directly from Equation (1).
Then we have and therefore, we can conclude with the theorem: Theorem 1. The sequence {g k (x)} k∈N satisfies the linear recurrence relation

Properties of the Basic Generating Function
We consider now a few properties of the basic generating functions G α (t, x). According to the definition (4a), the polynomials p k (x) are recognized as associated Sheffer polynomials [10] and quasi-monomials, according to the Dattoli [11,12] definition.

Differential Equation
We have: where and its functional inverse is given by so that, recalling the results by Y. Ben Cheikh [13], we find the derivative and multiplication operators of the quasi-monomials p k (x), in the form: and we can conclude that Theorem 2. The polynomials p k (x) satisfy the differential equation: that is, ∀n ≥ 1:

Differential Identity
Differentiating Equation (11) with respect to x, we find so that we can conclude with the theorem: The polynomials p k (x) satisfy the differential identity:

Extension by Convolution
We now consider the case of a generating function of the type: A straightforward consequence is the convolution formula for the resulting polynomials: so that the q h (x; c; a, b) can be found recursively by solving the infinite system Noting that p 0 (x; a, b) = g 0 (x; c; a, b) = 1, the very first polynomials are given by Further values can be obtained by using symbolic computation.

The General Recurrence Relation
From Equation (17) we find:

Extension to the General Case
We now generalize the convolution formula in Section 3.2, putting for shortness [c] r−1 = c 1 , c 2 , . . . , c r−1 , [a] r = a 1 , a 2 , . . . , a r , and considering the generating function: so that we find the convolution formula:

Extension to Matrix Polynomials
Extensions to Matrix polynomials have become a fashionable subject recently (see e.g., [22] and the references therein).
Since all powers of a matrix A commute, even every matrix polynomial commute. More generally, if σ(A) ⊂ Ω, where Ω is an open set of the complex plane, for any holomorphic functions f and g, this results in: that is, the involved matrix functions commute.

Conclusions
Starting from the results by Chen Zhuoyu and Qi Lan [9], we have shown convolution formulas and linear recurrence relations satisfied by a generating function containing several parameters. This can be used for number sequences (assuming x = 1) or polynomial sequences, depending on several parameters. Illustrative examples are shown both in case of second order or high order recurrence relations.
An extension to the case of matrix polynomials is also included.

Author Contributions:
The authors claim to have contributed equally and significantly in this paper. Both authors read and approved the final manuscript.
Funding: This research received no external funding.