Some notes on the algebraic structure of linear recurrent sequences

Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products are R-algebras, given any commutative ring $R$ with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these $R-$algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.


Introduction
Given a commutative ring with identity R, we will denote by S(R) the set of all sequences a = (a n ) n≥0 such that a n ∈ R, for all n ∈ N. A sequence a ∈ S(R) is said to be a linear recurrent sequence with characteristic polynomial p a (t) = t N − N −1 i=0 h i t N −i if its elements satisfy the following relation for all n ≥ N and the elements a 0 , . . .a N −1 are called initial conditions.We will denote by W(R) ⊂ S(R) the set of all linear recurrent sequences.Moreover, given a ∈ S(R), we will write A o (t) = ∞ n=0 a n t n for the ordinary generating function (o.g.f.) and A e (t) = ∞ n=0 a n n! t n for the exponential generating function (e.g.f.).It is well known that S(R) and W(R) can be equipped with several operations giving them interesting algebraic structures.When R is a field, it is immediate to see that the element-wise sum or product (also called the Hadamard product) of two linear recurrent sequences is still a linear recurrent sequence, see, e.g., [8].In [7], the authors proved it in the more general case where R is a ring, showing that W(R) is an R−algebra and giving also explicitly the characteristic polynomials of the elementwise sum and Hadamard product of two linear recurrent sequences.Larson and Taft [17,23] studied this algebraic structure characterizing the invertible elements and zero divisors.Further studies about the behaviour of linear recurrent sequences under the Hadamard product can be found, e.g., in [6,10,12,24].Similarly, W(R) equipped with the element-wise sum and the convolution product (or Cauchy product) has been deeply studied.For instance, W(R) is still an R−algebra and the characteristic polynomial of the convolution product between two linear recurrent sequences can be explicitly found [7].The convolution product of linear recurrent sequences is very important in many applications and it has been studied also from a combinatorial point of view [1] and over finite fields [11].For other results, see, e. g., [20,21,22].Another important operation between sequences is the binomial convolution (or Hurwitz product).The Hurwitz series ring, introduced in a systematic way by Keigher [13], has been extensively studied by several authors [2,3,4,5,14,18].However, there are few results when focusing on linear recurrent sequences [15,16].
In this paper, we extend the studies about the algebraic structure of linear recurrent sequences considering in particular the Hurwitz product and the Newton product (which is the generalization of the Hurwitz product considering multinomial coefficients).In particular, we prove that W(R) is an R−algebra when equipped with the element-wise sum and the Hurwitz product, as well as when we consider element-wise sum and Newton product.We also give explicitly the characteristic polynomials of the Hurwitz and Newton product of two linear recurrent sequences.For the Newton product we find explicitly also the inverses.Moreover, we study the isomorphisms between these algebraic structures, finding that W(R) with element-wise sum and Hurwitz product is not isomorphic to the other algebraic structures, whereas if we consider the Newton product, there is an isomorphism with the R−algebra obtained using the Hadamard product.Finally, we provide an overview about the behaviour of linear recurrent sequences under all the different operations considered (elementwise sum, Hadamard product, Cauchy product, Hurwitz product, Newton product) with respect to the characteristic polynomials and their companion matrices.

Preliminaries and notation
For any a, b ∈ S(R), we will deal with the following operations: • componentwise sum +, defined by • componentwise product or Hadamard product ⊙, defined by • convolution product * , defined by • binomial convolution product or Hurwitz product ⋆, defined by • multinomial convolution product or Newton product ⊠, defined by Remark 1.The Newton product is also called multinomial convolution product, since it is the natural generalization of the binomial convolution product using the multinomial coefficient, observing that In [7], the authors showed that (W(R), +, ⊙) and (W(R), +, * ) are R−algebras and they are never isomorphic.Moreover, given a, b ∈ W(R) and c = a ⊙ b, d = a * b, they proved that where the operation ⊗ between polynomials is defined as follows.Given two polynomials f (t) and g(t) with coefficients in R, said F and G their companion matrices, respectively, then f (t) ⊗ g(t) is the characteristic polynomial of the Kronecker product between F and G.In the following, we will denote by ⊗ also the Kronecker product between matrices.To the best of our knowledge, similar results involving the Hurwitz product and the Newton product are still missing.
Remark 2. Let us observe that the sequences c and d, defined above, recur with characteristic polynomials p c (t) and p d (t) as given in (1), respectively, but these polynomials are not necessarily the minimal polynomials of recurrence.Indeed, it is an hard problem to find the minimal polynomials of recurrence of these sequences, for some results, see [6,10,17,20].Proof.See [7].Definition 4. Given two monic polynomials f (t) and g(t) of degree M and N, respectively, their resultant is res(f (t), g(t)) := M i=1 N j=1 (α i − β j ), where α i 's and β j 's are the roots of f (t) and g(t), respectively.

R-algebras of linear recurrent sequences
Theorem 5. Given a, b ∈ W(R), we have that r = a ⋆ b ∈ W(R) and the characteristic polynomial of r is res(p a (x), p b (t−x)) with p b (t−x) regarded as a polynomial in t.Moreover, (W(R), +, ⋆) is an R−algebra.
Proof.It is well-known that (S(R), +, ⋆) is an R−algebra (see, e.g., [13]), thus it is sufficient to show that W(R) is closed under the Hurwitz product for proving that (W(R), +, ⋆) is an R−algebra.
Let M and N be the degrees of p a (t) and p b (t), respectively.Let us suppose p a (t) and p b (t) have distinct roots denoted by α 1 , . . ., α M and β 1 , . . ., β N , respectively.We consider the ordinary generating function of the sequence r = a ⋆ b, where b m t m is the Hadamard product between the sequence b and is the ordinary generating function of the linear recurrent sequence b, it is a rational function and we can write it as for some integers c j 's.Now, we have and we get that Thus, from (2) we obtain In particular, it is possible to rearrange the last formula in the following way Combining ( 3) and ( 4) we get Moreover the function δ t 1−β j t can be written in the following way Since the degree of the polynomial ( 6) is less than or equal to MN − 1, by Lemma 3, then r is a linear recurrent sequence and p(t) = res(p a (x), p b (t − x)) is its characteristic polynomial, as desired.
From the definition of Newton product, we want to prove the following equality Exploiting the Newton's inversion formula, i.e., for some arithmetic functions f and g, equation (7) becomes The previous proposition can be proved also exploiting the umbral calculus (see [19] for the basic notions).Given a, b ∈ S(R), let us consider two linear functionals U and V defined by and, applying the functionals U and V , it becomes Now, the last quantity can be rewritten as which is the n-th term of the sequence (a ⋆ 1) ⊙ (b ⋆ 1).Proof.Firstly, we show that (S(R), +, ⊠) is an R−algebra.This is an immediate consequence of Proposition 7. Indeed, since a ⊠ b = [(a ⋆ 1) ⊙ (b ⋆ 1)] ⋆ e, it is straightforward to see that ⊠ satisfies all the properties in order that (S(R), +, ⊠) is an R−algebra.Moreover, we can also see that (1, 0, 0, . ..) is the identity element for the Newton product.Indeed, it is sufficient to observe that (1, 0, 0, . ..) is the identity element for the Hurwitz product and e is the inverse of 1 with respect to ⋆.Then, it is immediate that (W(R), +, ⊠) is also an R−algebra, since given a, b ∈ W(R), we have a ⊠ b ∈ W(R) by Proposition 7.
Proposition 10.Given a ∈ S(R), said b its inverse with respect to the Newton product, then for any n ≥ 0.
Proof.Remembering that the identity element for the Newton product is (1, 0, 0, . ..), we have that a 0 b 0 must be 1, i.e., b 0 = a −1 0 .When n ≥ 1, we have that , applying Newton's inversion formula, we get and finally from which the thesis follows.
Remark 11.Let us observe that a is invertible with respect to the Newton product if and only if all the elements of a are invertible elements of R, as well as it happens for the Hadamard product.
Let us observe that every a ∈ W(R) can be associated to its monic characteristic polynomial p a (t) with coefficients in R and this polynomial to its companion matrix A. As studied the R−algebras of kind (W(R), +, ⊙), (W(R), +, * ), (W(R), +, ⋆) and (W(R), +, ⊠), it is interesting to give to the set of the monic polynomials Pol(R) with coefficients in R some new algebraic structures.Moreover, we can also observe what happens to the roots and to the companion matrices of the characteristic polynomials.
Let us consider a, b ∈ W(R) with characteristic polynomials of degree M and N, whose roots are α 1 , . . ., α M and β 1 , . . .β N , respectively.The sequences a + b and a * b both recur with characteristic polynomial p a (t) • p b (t).
Regarding the Hadamard product, we have already observed that the characteristic polynomial of c = a ⊙ b is p c (t) = p a (t) ⊗ p b (t), whose roots are α i β j , for i = 1, . . ., M and j = 1, . . ., N. Thus, starting from the R−algebra (W(R), +, ⊙), we can construct the semiring (Pol(R), •, ⊗) with identity the polynomial t − 1. Said A, B, and C the companion matrices of p a (t), p b (t) and p c (t), we have that C = A ⊗ B, where ⊗ is the Kronecker product between matrices.Thus C is a mn × mn matrix with eigenvalues the products of the eigenvalues of A and B.
Similarly, starting from the Hurwitz product, we can construct a new operation in Pol(R).Given c = a⋆b, we proved that p c (t) has roots α i +β j , for i = 1, . . ., M and j = 1, . . ., N. The matrix A ⊗ I n + I m ⊗ B is a mn × mn matrix, whose eigenvalues are the sum of the eigenvalues of A and B. Thus, we can define p c (t) = p a (t) ⋆ p b (t) as the characteristic polynomial of the matrix A ⊗ I n + I m ⊗ B and we get the semiring (Pol(R), •, ⋆).
Finally, given c = a ⊠ b, we know that p c (t) has roots α i + β j + α i β j , for i = 1, . . ., M and j = 1, . . ., N. In this case, we can define p c (t) = p a (t) ⊠ p b (t) as the characteristic polynomial of the matrix A ⊗ I n + I m ⊗ B + A ⊗ B, which is a mn × mn matrix, whose eigenvalues are exactly α i + β j + α i β j , for i = 1, . . ., M and j = 1, . . ., N. Thus, we have that (Pol(R), •, ⊠) is another semiring of monic polynomials.

On isomorphisms between R-algebras
In [7], the authors proved that (W(R), +, ⊙) and (W(R), +, * ) are never isomorphic as R−algebras.In the following we prove similar results for the other algebraic structures that we have studied in the previous section.

Theorem 9 .
Given a, b ∈ W(R), we have that c = a ⊠ b ∈ W(R) and the characteristic polynomial of c is M i=1 N j=1 (t − (α i + β j + α i β j )), where M = deg(p a (t)), N = deg(p b (t)), α i 's are the roots of p a (t) and β j 's the roots of p b (t).Moreover, (W(R), +, ⊠) is an R−algebra.

s
i s n−i t n ⇒ 2k+1 i=0 s i s 2k+1−i = 0 2k i=0 s i s 2k−i = s k 3. Given a ∈ S(R), we have that a ∈ W(R) if and only if p * a (t) • A o (t) is a polynomial of degree less than deg(p a (t)), where p * a (t) denotes the reciprocal or reflected polynomial of p a (t).