Some notes on the algebraic structure of linear recurrent sequences

Gessica Alecci; Stefano Barbero; Nadir Murru

arXiv:2302.13867·math.NT·February 28, 2023

Some notes on the algebraic structure of linear recurrent sequences

Gessica Alecci, Stefano Barbero, Nadir Murru

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TL;DR

This paper explores the algebraic structures of linear recurrent sequences under various operations, proving they form R-algebras and providing explicit characteristic polynomials, with an investigation into their isomorphisms.

Contribution

It establishes that sets of linear recurrent sequences form R-algebras under certain operations and explicitly characterizes their algebraic properties.

Findings

01

Linear recurrent sequences form R-algebras with sum and product operations.

02

Explicit characteristic polynomials for Hurwitz and Newton products are provided.

03

Investigation into isomorphisms between different R-algebras is conducted.

Abstract

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.

Equations104

a_{n} = i = 0 \sum N - 1 h_{i} a_{n - i}

a_{n} = i = 0 \sum N - 1 h_{i} a_{n - i}

a + b = c, c_{n} = a_{n} + b_{n}, \forall n \geq 0;

a + b = c, c_{n} = a_{n} + b_{n}, \forall n \geq 0;

a ⊙ b = c, c_{n} = a_{n} \cdot b_{n}, \forall n \geq 0;

a ⊙ b = c, c_{n} = a_{n} \cdot b_{n}, \forall n \geq 0;

a * b = c, c_{n} = i = 0 \sum n a_{i} b_{n - i}, \forall n \geq 0;

a * b = c, c_{n} = i = 0 \sum n a_{i} b_{n - i}, \forall n \geq 0;

a ⋆ b = c, c_{n} = i = 0 \sum n (i n) a_{i} b_{n - i}, \forall n \geq 0;

a ⋆ b = c, c_{n} = i = 0 \sum n (i n) a_{i} b_{n - i}, \forall n \geq 0;

a ⊠ b = c, c_{n} = i = 0 \sum n j = 0 \sum i (i n) (j i) a_{i} b_{n - j}, \forall n \geq 0.

a ⊠ b = c, c_{n} = i = 0 \sum n j = 0 \sum i (i n) (j i) a_{i} b_{n - j}, \forall n \geq 0.

p_{c} (t) = p_{a} (t) \otimes p_{b} (t), p_{d} (t) = p_{a} (t) \cdot p_{b} (t),

p_{c} (t) = p_{a} (t) \otimes p_{b} (t), p_{d} (t) = p_{a} (t) \cdot p_{b} (t),

R_{o} (t) = n = 0 \sum + \infty (i = 0 \sum n (i n) a_{i} b_{n - i}) t^{n}

R_{o} (t) = n = 0 \sum + \infty (i = 0 \sum n (i n) a_{i} b_{n - i}) t^{n}

= i = 0 \sum + \infty a_{i} t^{i} n = i \sum + \infty (i n) b_{n - i} t^{n - i}

= i = 0 \sum + \infty a_{i} t^{i} m = 0 \sum + \infty (i m + i) b_{m} t^{m}

B_{o} (t) = \frac{γ ( t )}{p _{b}^{*} ( t )} = j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t )},

B_{o} (t) = \frac{γ ( t )}{p _{b}^{*} ( t )} = j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t )},

\frac{1}{1 - β _{j} t} ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = m = 0 \sum + \infty (b_{j} t)^{m} ⊙ m = 0 \sum + \infty (i m + i) t^{m} = m = 0 \sum + \infty (i m + i) (b_{j} t)^{m} = \frac{1}{( 1 - β _{j} t ) ^{i + 1}}

\frac{1}{1 - β _{j} t} ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = m = 0 \sum + \infty (b_{j} t)^{m} ⊙ m = 0 \sum + \infty (i m + i) t^{m} = m = 0 \sum + \infty (i m + i) (b_{j} t)^{m} = \frac{1}{( 1 - β _{j} t ) ^{i + 1}}

B_{o} (t) ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = j = 1 \sum N c_{j} \frac{1}{1 - β _{j} t} ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t ) ^{i + 1}} .

B_{o} (t) ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = j = 1 \sum N c_{j} \frac{1}{1 - β _{j} t} ⊙ \frac{1}{( 1 - t ) ^{i + 1}} = j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t ) ^{i + 1}} .

R_{o} (t) = i = 0 \sum + \infty a_{i} t^{i} j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t ) ^{i + 1}}

R_{o} (t) = i = 0 \sum + \infty a_{i} t^{i} j = 1 \sum N \frac{c _{j}}{( 1 - β _{j} t ) ^{i + 1}}

= j = 1 \sum N \frac{c _{j}}{1 - β _{j} t} A_{o} (\frac{t}{1 - β _{j} t})

= j = 1 \sum N \frac{c _{j}}{1 - β _{j} t} \frac{δ ( \frac{t}{1 - β _{j} t} )}{p _{a}^{*} ( \frac{t}{1 - β _{j} t} )}

p^{*} (t) = h = 1 \prod M l = 1 \prod N (1 - β_{l} t - α_{h} t)

p^{*} (t) = h = 1 \prod M l = 1 \prod N (1 - β_{l} t - α_{h} t)

= h = 1 \prod M l = 1 \prod N (1 - β_{l} t) l = 1 \prod N (1 - \frac{α _{h} t}{1 - β _{l} t})

= h = 1 \prod M p_{b}^{*} (t) l = 1 \prod N (1 - \frac{α _{h} t}{1 - β _{l} t})

= [p_{b}^{*} (t)]^{M} l = 1 \prod N h = 1 \prod M (1 - \frac{α _{h} t}{1 - β _{l} t})

= [p_{b}^{*} (t)]^{M} l = 1 \prod N p_{a}^{*} (\frac{t}{1 - β _{l} t})

p^{*} (t) \cdot R_{o} (t)

p^{*} (t) \cdot R_{o} (t)

= [p_{b}^{*} (t)]^{M} (j = 1 \sum N \frac{c _{j}}{1 - β _{j} t} \cdot δ (\frac{t}{1 - β _{j} t})) \cdot l = 1 l \neq = j \prod N p_{a}^{*} (\frac{t}{1 - β _{l} t})

δ (\frac{t}{1 - β _{j} t}) = h = 0 \sum M - 1 δ_{h} \cdot (\frac{t}{1 - β _{j} t})^{h} = \frac{\sum _{h = 0}^{M - 1} δ _{h} t ^{h} ( 1 - β _{j} t ) ^{M - 1 - h}}{( 1 - β _{j} t ) ^{M - 1}} = \frac{μ _{j} ( t )}{( 1 - β _{j} t ) ^{M - 1}}

δ (\frac{t}{1 - β _{j} t}) = h = 0 \sum M - 1 δ_{h} \cdot (\frac{t}{1 - β _{j} t})^{h} = \frac{\sum _{h = 0}^{M - 1} δ _{h} t ^{h} ( 1 - β _{j} t ) ^{M - 1 - h}}{( 1 - β _{j} t ) ^{M - 1}} = \frac{μ _{j} ( t )}{( 1 - β _{j} t ) ^{M - 1}}

p_{a}^{*} (\frac{t}{1 - β _{j} t}) = \frac{\sum _{h = 0}^{M} f _{h} t ^{h} ( 1 - β _{j} t ) ^{M - h}}{( 1 - β _{j} t ) ^{M}} = \frac{ξ _{j} ( t )}{( 1 - β _{j} t ) ^{M}}

p_{a}^{*} (\frac{t}{1 - β _{j} t}) = \frac{\sum _{h = 0}^{M} f _{h} t ^{h} ( 1 - β _{j} t ) ^{M - h}}{( 1 - β _{j} t ) ^{M}} = \frac{ξ _{j} ( t )}{( 1 - β _{j} t ) ^{M}}

[p_{b}^{*} (t)]^{M} j = 1 \sum N \frac{c _{j}}{1 - β _{j} t} \cdot \frac{μ _{j} ( t )}{( 1 - β _{j} t ) ^{M - 1}} \cdot l = 1 l \neq = j \prod N \frac{ξ _{j} ( t )}{( 1 - β _{j} t ) ^{M}}

[p_{b}^{*} (t)]^{M} j = 1 \sum N \frac{c _{j}}{1 - β _{j} t} \cdot \frac{μ _{j} ( t )}{( 1 - β _{j} t ) ^{M - 1}} \cdot l = 1 l \neq = j \prod N \frac{ξ _{j} ( t )}{( 1 - β _{j} t ) ^{M}}

= [p_{b}^{*} (t)]^{M} j = 1 \sum N c_{j} \cdot \frac{μ _{j} ( t )}{( 1 - β _{j} t ) ^{M}} \frac{l = 1 l \neq = j \prod N ξ _{j} ( t )}{l = 1 l \neq = j \prod N ( 1 - β _{j} t ) ^{M}}

= [p_{b}^{*} (t)]^{M} j = 1 \sum N c_{j} μ_{j} (t) \frac{l = 1 l \neq = j \prod N ξ _{j} ( t )}{[ p _{b}^{*} ( t ) ] ^{M}} = j = 1 \sum N c_{j} μ_{j} (t) l = 1 l \neq = j \prod N ξ_{j} (t) .

i = 0 \sum n (i n) (- 1)^{n - i} s = 0 \sum i (s i) a_{s} t = 0 \sum i (t i) b_{t} = i = 0 \sum n j = 0 \sum i (i n) (j i) a_{i} b_{n - j}, \forall n \geq 0.

i = 0 \sum n (i n) (- 1)^{n - i} s = 0 \sum i (s i) a_{s} t = 0 \sum i (t i) b_{t} = i = 0 \sum n j = 0 \sum i (i n) (j i) a_{i} b_{n - j}, \forall n \geq 0.

i = 0 \sum n (i n) (- 1)^{i} c_{i} = (- 1)^{n} d_{n} .

i = 0 \sum n (i n) (- 1)^{i} c_{i} = (- 1)^{n} d_{n} .

i = 0 \sum n (i n) (- 1)^{i} f (i) = g (n) \Leftrightarrow f (n) = i = 0 \sum n (i n) (- 1)^{i} g (i),

i = 0 \sum n (i n) (- 1)^{i} f (i) = g (n) \Leftrightarrow f (n) = i = 0 \sum n (i n) (- 1)^{i} g (i),

i = 0 \sum n (i n) (- 1)^{i} (- 1)^{i} d_{i} = c_{n}

i = 0 \sum n (i n) (- 1)^{i} (- 1)^{i} d_{i} = c_{n}

i = 0 \sum n (i n) k = 0 \sum i j = 0 \sum k (k i) (j k) a_{k} b_{i - j} = s = 0 \sum n (s n) a_{s} t = 0 \sum n (t n) b_{t} .

i = 0 \sum n (i n) k = 0 \sum i j = 0 \sum k (k i) (j k) a_{k} b_{i - j} = s = 0 \sum n (s n) a_{s} t = 0 \sum n (t n) b_{t} .

j = 0 \sum n s = j \sum n i = s \sum n (i n) (s i) (j s) a_{s} b_{i - j}

j = 0 \sum n s = j \sum n i = s \sum n (i n) (s i) (j s) a_{s} b_{i - j}

= s = 0 \sum n (s n) a_{s} j = 0 \sum s t = s - j \sum n - j (n - t - j n - s) (j s) b_{t}

= s = 0 \sum n (s n) a_{s} m = 0 \sum s t = m \sum n - s + m (n - s - ( t - m ) n - s) (s - m s) b_{t}

= s = 0 \sum n (s n) a_{s} m = 0 \sum s t = m \sum n - s + m (t - m n - s) (m s) b_{t}

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Full text

Some notes on the algebraic structure of linear recurrent sequences

Gessica Alecci1, Stefano Barbero1, Nadir Murru2

1 Department of Mathematical Sciences G. L. Lagrange, Politecnico di Torino

[email protected], [email protected]

2 Department of Mathematics, Università di Trento

[email protected]

Abstract.

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.

1. Introduction

Given a commutative ring with identity $R$ , we will denote by $\mathcal{S}(R)$ the set of all sequences $\mathbf{a}=(a_{n})_{n\geq 0}$ such that $a_{n}\in R$ , for all $n\in\mathbb{N}$ . A sequence $\mathbf{a}\in\mathcal{S}(R)$ is said to be a linear recurrent sequence with characteristic polynomial $p_{a}(t)=t^{N}-\sum_{i=0}^{N-1}h_{i}t^{N-i}$ if its elements satisfy the following relation

[TABLE]

for all $n\geq N$ and the elements $a_{0},\ldots a_{N-1}$ are called initial conditions. We will denote by $\mathcal{W}(R)\subset\mathcal{S}(R)$ the set of all linear recurrent sequences. Moreover, given $\mathbf{a}\in\mathcal{S}(R)$ , we will write $A_{o}(t)=\sum_{n=0}^{\infty}a_{n}t^{n}$ for the ordinary generating function (o.g.f.) and $A_{e}(t)=\sum_{n=0}^{\infty}\cfrac{a_{n}}{n!}t^{n}$ for the exponential generating function (e.g.f.). It is well known that $\mathcal{S}(R)$ and $\mathcal{W}(R)$ 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 $\mathcal{W}(R)$ is an $R-$ algebra and giving also explicitly the characteristic polynomials of the element-wise 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, $\mathcal{W}(R)$ equipped with the element-wise sum and the convolution product (or Cauchy product) has been deeply studied. For instance, $\mathcal{W}(R)$ 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 $\mathcal{W}(R)$ 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 $\mathcal{W}(R)$ 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 (element-wise sum, Hadamard product, Cauchy product, Hurwitz product, Newton product) with respect to the characteristic polynomials and their companion matrices.

2. Preliminaries and notation

For any $\mathbf{a},\mathbf{b}\in\mathcal{S}(R)$ , we will deal with the following operations:

•

componentwise sum $+$ , defined by

[TABLE]

•

componentwise product or Hadamard product $\odot$ , defined by

[TABLE]

•

convolution product $*$ , defined by

[TABLE]

•

binomial convolution product or Hurwitz product $\star$ , defined by

[TABLE]

•

multinomial convolution product or Newton product $\boxtimes$ , defined by

[TABLE]

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 $\binom{n}{i}\binom{i}{j}=\binom{n}{n-i,i-j,j}$ .

In [7], the authors showed that $(\mathcal{W}(R),+,\odot)$ and $(\mathcal{W}(R),+,*)$ are $R-$ algebras and they are never isomorphic. Moreover, given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ and $\mathbf{c}=\mathbf{a}\odot\mathbf{b}$ , $\mathbf{d}=\mathbf{a}*\mathbf{b}$ , they proved that

[TABLE]

where the operation $\otimes$ 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 $f(t)\otimes g(t)$ is the characteristic polynomial of the Kronecker product between $F$ and $G$ . In the following, we will denote by $\otimes$ 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 $\mathbf{c}$ and $\mathbf{d}$ , 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].

Lemma 3.

Given $\mathbf{a}\in\mathcal{S}(R)$ , we have that $\mathbf{a}\in\mathcal{W}(R)$ if and only if $p^{*}_{a}(t)\cdot A_{o}(t)$ is a polynomial of degree less than $\deg(p_{a}(t))$ , where $p^{*}_{a}(t)$ denotes the reciprocal or reflected polynomial of $p_{a}(t)$ .

Proof.

See [7]. ∎

Definition 4.

Given two monic polynomials $f(t)$ and $g(t)$ of degree $M$ and $N$ , respectively, their resultant is $\text{res}(f(t),g(t)):=\prod_{i=1}^{M}\prod_{j=1}^{N}(\alpha_{i}-\beta_{j})$ , where $\alpha_{i}$ ’s and $\beta_{j}$ ’s are the roots of $f(t)$ and $g(t)$ , respectively.

3. R-algebras of linear recurrent sequences

Theorem 5.

Given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , we have that $\mathbf{r}=\mathbf{a}\star\mathbf{b}\in\mathcal{W}(R)$ and the characteristic polynomial of $\mathbf{r}$ is $\text{res}(p_{a}(x),p_{b}(t-x))$ with $p_{b}(t-x)$ regarded as a polynomial in t. Moreover, $(\mathcal{W}(R),+,\star)$ is an $R-$ algebra.

Proof.

It is well–known that $(\mathcal{S}(R),+,\star)$ is an $R-$ algebra (see, e.g., [13]), thus it is sufficient to show that $\mathcal{W}(R)$ is closed under the Hurwitz product for proving that $(\mathcal{W}(R),+,\star)$ 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 $\alpha_{1},\ldots,\alpha_{M}$ and $\beta_{1},\ldots,\beta_{N}$ , respectively. We consider the ordinary generating function of the sequence $\mathbf{r}=\mathbf{a}\star\mathbf{b}$ ,

[TABLE]

where $\sum_{m=0}^{+\infty}\binom{m+i}{i}b_{m}t^{m}$ is the Hadamard product between the sequence $\mathbf{b}$ and $\left(\binom{m+i}{i}\right)_{m\geq 0}$ , i.e., $\left(\sum_{m=0}^{+\infty}b_{m}t^{m}\right)\odot\left(\sum_{m=0}^{+\infty}\binom{m+i}{i}t^{m}\right)=B_{o}(t)\odot\cfrac{1}{{(1-t)}^{i+1}}$ . Since $B_{o}(t)$ is the ordinary generating function of the linear recurrent sequence $\mathbf{b}$ , it is a rational function and we can write it as

[TABLE]

for some integers $c_{j}$ ’s. Now, we have

[TABLE]

and we get that

[TABLE]

Thus, from (3) we obtain

[TABLE]

Let $p(t)=\text{res}(p_{a}(x),p_{b}(t-x))$ , then ${\displaystyle p(t)=\prod_{h=1}^{M}\prod_{l=1}^{N}(t-\alpha_{h}-\beta_{l}})$ and its reciprocal polynomial is $p^{*}(t)=\prod_{h=1}^{M}\prod_{l=1}^{N}(1-(\alpha_{h}+\beta_{l})t)$ . In particular, it is possible to rearrange the last formula in the following way

[TABLE]

Combining (3) and (3) we get

[TABLE]

Moreover the function $\delta\left(\frac{t}{1-\beta_{j}t}\right)$ can be written in the following way

[TABLE]

with $deg(\mu_{j}(t))\leq M-1$ . Applying the same reasoning,

[TABLE]

with $deg(\xi_{j}(t))\leq M$ . Hence, equation (3) becomes

[TABLE]

Since the degree of the polynomial (3) is less than or equal to $MN-1$ , by Lemma 3, then $\mathbf{r}$ is a linear recurrent sequence and $p(t)=\text{res}(p_{a}(x),p_{b}(t-x))$ is its characteristic polynomial, as desired. ∎

Remark 6.

Given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , if $\alpha_{1},\ldots\alpha_{M}$ and $\beta_{1},\ldots,\beta_{N}$ are distinct roots of $p_{a}(t)$ and $p_{b}(t)$ respectively, then, by Theorem 5, the roots of the characteristic polynomial of $\mathbf{a}\star\mathbf{b}$ are $\alpha_{i}+\beta_{j}$ , for any $i=1,\ldots,M$ and $j=1,\ldots,N$ . Moreover, we would like to highlight that the proof of Theorem 5 can be adapted also in the case of multiple roots.

Proposition 7.

Given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , then $\mathbf{a}\boxtimes\mathbf{b}=[(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})]\star\mathbf{e}$ , where $\mathbf{1}:=(1,1,1,\ldots)$ and $\mathbf{e}:=((-1)^{n})_{n\geq 0}$ .

Proof.

The $n$ -th terms of $\mathbf{a}\star\mathbf{1}$ and $\mathbf{b}\star\mathbf{1}$ are by definition $\sum_{i=0}^{n}\binom{n}{i}a_{i}$ and $\sum_{i=0}^{n}\binom{n}{i}b_{i}$ , respectively. Thus, the $n$ –th term of $(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})$ is $\sum_{s=0}^{n}\binom{n}{s}a_{s}\sum_{t=0}^{n}\binom{n}{t}b_{t}$ , and thus $\sum_{i=0}^{n}\binom{n}{i}(-1)^{n-i}\sum_{s=0}^{i}\binom{i}{s}a_{s}\sum_{t=0}^{i}\binom{i}{t}b_{t}$ is the $n$ –th term of $[(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})]\star\mathbf{e}$ .

From the definition of Newton product, we want to prove the following equality

[TABLE]

Let $c_{i}=\sum_{s=0}^{i}\binom{i}{s}a_{s}\sum_{t=0}^{i}\binom{i}{t}b_{t}$ and $d_{n}=\sum_{i=0}^{n}\sum_{j=0}^{i}\binom{n}{i}\binom{i}{j}a_{i}b_{n-j}$ , then the previous identity is equivalent to

[TABLE]

Exploiting the Newton’s inversion formula, i.e.,

[TABLE]

for some arithmetic functions $f$ and $g$ , equation (7) becomes

[TABLE]

that is

[TABLE]

Now, we can write the first member as

[TABLE]

where the last equivalence is due to the Vandermonde’s identity $\sum_{m=0}^{t}\binom{n-s}{t-m}\binom{s}{m}=\sum_{t=0}^{n}\binom{n}{t}$ . ∎

Remark 8.

The previous proposition can be proved also exploiting the umbral calculus (see [19] for the basic notions). Given $\mathbf{a},\mathbf{b}\in\mathcal{S}(R)$ , let us consider two linear functionals $U$ and $V$ defined by $U(W^{n})=a_{n}$ and $V(Z^{n})=b_{n}$ . The $n$ –th term of $(\mathbf{a}\boxtimes\mathbf{b})\star\mathbf{1}$ is $\sum_{i=0}^{n}\binom{n}{i}\sum_{k=0}^{i}\sum_{j=0}^{k}\binom{i}{k}\binom{k}{j}a_{k}b_{i-j}$ and, applying the functionals $U$ and $V$ , it becomes

[TABLE]

Now, the last quantity can be rewritten as

[TABLE]

which is the $n$ –th term of the sequence $(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})$ .

Theorem 9.

Given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , we have that $\mathbf{c}=\mathbf{a}\boxtimes\mathbf{b}\in\mathcal{W}(R)$ and the characteristic polynomial of $\mathbf{c}$ is $\prod_{i=1}^{M}\prod_{j=1}^{N}(t-(\alpha_{i}+\beta_{j}+\alpha_{i}\beta_{j}))$ , where $M=\deg(p_{a}(t))$ , $N=\deg(p_{b}(t))$ , $\alpha_{i}$ ’s are the roots of $p_{a}(t)$ and $\beta_{j}$ ’s the roots of $p_{b}(t)$ . Moreover, $(\mathcal{W}(R),+,\boxtimes)$ is an $R-$ algebra.

Proof.

Firstly, we show that $(\mathcal{S}(R),+,\boxtimes)$ is an $R-$ algebra. This is an immediate consequence of Proposition 7. Indeed, since $\mathbf{a}\boxtimes\mathbf{b}=[(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})]\star\mathbf{e}$ , it is straightforward to see that $\boxtimes$ satisfies all the properties in order that $(\mathcal{S}(R),+,\boxtimes)$ is an $R-$ algebra. Moreover, we can also see that $(1,0,0,\ldots)$ is the identity element for the Newton product. Indeed, it is sufficient to observe that $(1,0,0,\ldots)$ is the identity element for the Hurwitz product and $\mathbf{e}$ is the inverse of $\mathbf{1}$ with respect to $\star$ . Then, it is immediate that $(\mathcal{W}(R),+,\boxtimes)$ is also an $R-$ algebra, since given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , we have $\mathbf{a}\boxtimes\mathbf{b}\in\mathcal{W}(R)$ by Proposition 7.

By Theorem 5 and Remark 6, we can observe that given $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , then $\mathbf{a}\star\mathbf{1}$ and $\mathbf{b}\star\mathbf{1}$ are linear recurrent sequences whose characteristic polynomials have roots $\alpha_{i}+1$ and $\beta_{j}+1$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ , respectively. Moreover, since $\mathbf{e}$ is a linear recurrent sequence whose characteristic polynomial is $t+1$ , then $[(\mathbf{a}\star\mathbf{1})\odot(\mathbf{b}\star\mathbf{1})]\star\mathbf{e}$ has characteristic polynomial whose roots are $(\alpha_{i}+1)(\beta_{j}+1)-1=\alpha_{i}+\beta_{j}+\alpha_{i}\beta_{j}$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ . ∎

Proposition 10.

Given $\mathbf{a}\in\mathcal{S}(R)$ , said $\mathbf{b}$ its inverse with respect to the Newton product, then

[TABLE]

for any $n\geq 0$ .

Proof.

Remembering that the identity element for the Newton product is $(1,0,0,\ldots)$ , we have that $a_{0}b_{0}$ must be $1$ , i.e., $b_{0}=a_{0}^{-1}$ . When $n\geq 1$ , we have that

[TABLE]

and, then

[TABLE]

Let $d_{i}=\frac{1}{\sum_{s=0}^{i}\binom{i}{s}a_{s}}$ , applying Newton’s inversion formula, we get

[TABLE]

and finally

[TABLE]

from which the thesis follows. ∎

Remark 11.

Let us observe that $\mathbf{a}$ is invertible with respect to the Newton product if and only if all the elements of $\mathbf{a}$ are invertible elements of $R$ , as well as it happens for the Hadamard product.

Let us observe that every $\mathbf{a}\in\mathcal{W}(R)$ 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 $(\mathcal{W}(R),+,\odot)$ , $(\mathcal{W}(R),+,*)$ , $(\mathcal{W}(R),+,\star)$ and $(\mathcal{W}(R),+,\boxtimes)$ , it is interesting to give to the set of the monic polynomials $\mathcal{P}ol(R)$ 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 $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ with characteristic polynomials of degree $M$ and $N$ , whose roots are $\alpha_{1},\ldots,\alpha_{M}$ and $\beta_{1},\ldots\beta_{N}$ , respectively. The sequences $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}*\mathbf{b}$ both recur with characteristic polynomial $p_{a}(t)\cdot p_{b}(t)$ .

Regarding the Hadamard product, we have already observed that the characteristic polynomial of $\mathbf{c}=\mathbf{a}\odot\mathbf{b}$ is $p_{c}(t)=p_{a}(t)\otimes p_{b}(t)$ , whose roots are $\alpha_{i}\beta_{j}$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ . Thus, starting from the $R-$ algebra $(\mathcal{W}(R),+,\odot)$ , we can construct the semiring $(\mathcal{P}ol(R),\cdot,\otimes)$ 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 $C=A\otimes B$ , where $\otimes$ is the Kronecker product between matrices. Thus $C$ is a $mn\times mn$ 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 $\mathcal{P}ol(R)$ . Given $\mathbf{c}=\mathbf{a}\star\mathbf{b}$ , we proved that $p_{c}(t)$ has roots $\alpha_{i}+\beta_{j}$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ . The matrix $A\otimes I_{n}+I_{m}\otimes B$ is a $mn\times mn$ matrix, whose eigenvalues are the sum of the eigenvalues of $A$ and $B$ . Thus, we can define $p_{c}(t)=p_{a}(t)\star p_{b}(t)$ as the characteristic polynomial of the matrix $A\otimes I_{n}+I_{m}\otimes B$ and we get the semiring $(\mathcal{P}ol(R),\cdot,\star)$ .

Finally, given $\mathbf{c}=\mathbf{a}\boxtimes\mathbf{b}$ , we know that $p_{c}(t)$ has roots $\alpha_{i}+\beta_{j}+\alpha_{i}\beta_{j}$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ . In this case, we can define $p_{c}(t)=p_{a}(t)\boxtimes p_{b}(t)$ as the characteristic polynomial of the matrix $A\otimes I_{n}+I_{m}\otimes B+A\otimes B$ , which is a $mn\times mn$ matrix, whose eigenvalues are exactly $\alpha_{i}+\beta_{j}+\alpha_{i}\beta_{j}$ , for $i=1,\ldots,M$ and $j=1,\ldots,N$ . Thus, we have that $(\mathcal{P}ol(R),\cdot,\boxtimes)$ is another semiring of monic polynomials.

4. On isomorphisms between R-algebras

In [7], the authors proved that $(\mathcal{W}(R),+,\odot)$ and $(\mathcal{W}(R),+,*)$ 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 12.

The $R-$ algebras $(\mathcal{W}(R),+,\odot)$ and $(\mathcal{W}(R),+,\star)$ are not isomorphic.

Proof.

Let us suppose that $\psi:(\mathcal{W}(R),+,\odot)\xrightarrow{}(\mathcal{W}(R),+,\star)$ is an injective morphism and consider $\mathbf{a}:=(1,0,0,0,\dots)$ and $\mathbf{b}:=(0,1,0,0,\dots)$ .

Then $\psi(\mathbf{a}\odot\mathbf{b})=\psi(\mathbf{a})\star\psi(\mathbf{b})=(0,0,\ldots)$ and, by injectivity, $\psi(\mathbf{a}),\psi(\mathbf{b})\neq(0,0,\ldots)$ . Let $A_{e}^{\psi}(t)$ and $B_{e}^{\psi}(t)$ be the exponential generating functions of $\psi(\mathbf{a})$ and $\psi(\mathbf{b})$ , respectively. From $\psi(\mathbf{a})\star\psi(\mathbf{b})=(0,0,\ldots)$ , it follows that $A_{e}^{\psi}(t)B_{e}^{\psi}(t)=0$ . From Lemma 3, we have

[TABLE]

with $deg(h(t))<deg(p^{*}_{\psi(\mathbf{a})}(t))$ .

Now, consider the map $\gamma:\sum_{n=0}^{\infty}a_{n}t^{n}\xrightarrow{}\sum_{n=0}^{\infty}\frac{a_{n}}{n!}t^{n}$ , which is an isomorphism between the ring of ordinary series and the Hurwitz series ring, see, e.g., [3]. Applying $\gamma$ to (9), we obtain

[TABLE]

where $p^{*}_{\psi(\mathbf{a})}(t)$ can be viewed as a formal series with an infinite number of zero coefficients. Multiplying by $\gamma(p^{*}_{\psi(\mathbf{a})}(t))$ the equation $A_{e}^{\psi}(t)B_{e}^{\psi}(t)=0$ , it becomes

[TABLE]

which implies $h(t)B_{o}^{\psi}(t)=0$ . From this, it follows that there is a nonzero element $w\in R$ , such that $wB_{o}^{\psi}(t)=0$ ([9, Eq. (2.9)]) and $w\mathbf{b}=0$ , which is absurd. ∎

Theorem 13.

The $R-$ algebras $(\mathcal{W}(R),+,\odot)$ and $(\mathcal{W}(R),+,\boxtimes)$ are isomorphic.

Proof.

The explicit isomorphism is $\psi:(\mathcal{W}(R),+,\odot)\rightarrow(\mathcal{W}(R),+,\boxtimes)$ defined by $\psi(\mathbf{a}):=\mathbf{a}\star\mathbf{e}$ , where $\mathbf{e}=((-1)^{n})_{n\geq 0}$ . Indeed, by Theorem 5, the map $\psi$ is well–defined (in the sense that a linear recurrent sequence is mapped into a linear recurrent sequence). Moreover, since $\mathbf{1}$ is the inverse of $\mathbf{e}$ with respect to the Hurwitz product, it is straightforward to check injectivity and surjectivity. Finally, by Proposition 7, we have

[TABLE]

that is equal to

[TABLE]

since $\mathbf{e}\star\mathbf{1}=(1,0,0,\ldots)$ . ∎

Theorem 14.

Let $R$ be an integral domain, if $\psi:(\mathcal{W}(R),+,*)\rightarrow(\mathcal{W}(R),+,\star)$ is a morphism, then $\psi$ is not injective.

Proof.

Let us suppose that $\psi:(\mathcal{W}(R),+,*)\rightarrow(\mathcal{W}(R),+,\star)$ is an injective morphism. Let us denote by $(\psi(\mathbf{a}))_{i}$ the $i$ -th term of the sequence $\psi(\mathbf{a})$ . The $n$ -th term of $\psi(\mathbf{a})\star\psi(\mathbf{b})$ is

[TABLE]

Then, considering $\psi(\mathbf{a}*\mathbf{b})=\psi(\mathbf{a})\star\psi(\mathbf{b})$ , for any $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ , we obtain

[TABLE]

where we define the formal sequences $\mathbf{f}:=(1,2!,3!,\ldots)$ and $\mathbf{f}^{-1}:=(1,\frac{1}{2!},\frac{1}{3!},\ldots)$ .

We define a map $\tau:(\mathcal{W}(R),+,*)\xrightarrow{}(\mathcal{W}(R),+,*)$ such that $\tau(\mathbf{a})=\psi(\mathbf{a})\odot\mathbf{f}^{-1}$ . By definition, we have that $\tau(\mathbf{a}+\mathbf{b})=\tau(\mathbf{a})+\tau(\mathbf{b})$ , $\tau(\mathbf{a}*\mathbf{b})=\tau(\mathbf{a})*\tau(\mathbf{b})$ , for any $\mathbf{a},\mathbf{b}\in\mathcal{W}(R)$ and $\ker(\tau)=\mathbf{0}$ where $\mathbf{0}=(0,0,0,\ldots)$ .

Let $\tilde{\tau}$ be a map that acts over the ordinary generating functions such that if $A(t)=\sum_{n=0}^{+\infty}a_{n}t^{n}$ then $\tilde{\tau}(A(t))=\sum_{n=0}^{+\infty}(\tau(a))_{n}t^{n}$ . From the properties of $\tau$ , it follows that $\tilde{\tau}(A(t)+B(t))=\tilde{\tau}(A(t))+\tilde{\tau}(B(t))$ , $\tilde{\tau}(A(t)B(t))=\tilde{\tau}(A(t))\tilde{\tau}(B(t))$ and $\tilde{\tau}(A(t))=\tilde{\tau}(B(t))\Leftrightarrow A(t)=B(t)$ .

Now, if we consider $A(t)=1$ and $B(t)=1$ , we clearly have $\tilde{\tau}(1)=\tilde{\tau}(1\cdot 1)=\tilde{\tau}(1)\tilde{\tau}(1)$ and this implies $\tilde{\tau}(1)=1$ . Indeed, by the injectivity of $\tau$ , we can not have $\tilde{\tau}(1)=0$ because $\tau(1,0,\ldots)$ should be $\mathbf{0}$ .

In the case that $A(t)=t$ and $B(t)=-t$ , then $0=\tilde{\tau}(0)=\tilde{\tau}(t-t)=\tilde{\tau}(t)+\tilde{\tau}(-t)$ , so $\tilde{\tau}(-t)=-\tilde{\tau}(t)$ . Moreover, when $A(t)=t$ and $B(t)=t^{-1}$ , then $1=\tilde{\tau}(1)=\tilde{\tau}(t\cdot t^{-1})=\tilde{\tau}(t)\tilde{\tau}(t^{-1})$ and $\tilde{\tau}(t^{-1})=(\tilde{\tau}(t))^{-1}$ .

Lastly, if $A(t)=B(t)=t$ , then $\tilde{\tau}(t^{2})=\tilde{\tau}(t\cdot t)=\tilde{\tau}(t)\tilde{\tau}(t)$ , so $\tilde{\tau}(t^{2})=(\tilde{\tau}(t))^{2}$ and $\tilde{\tau}(nt)=n\tilde{\tau}(t)$ , $\forall n\in\mathbb{N}$ .

Let us consider $\tilde{\tau}(t)=\sum_{n=0}^{+\infty}s_{n}t^{n}$ , then from $\tilde{\tau}(t^{2})=(\tilde{\tau}(t))^{2}$ it follows that

[TABLE]

From (11), we obtain $s_{0}=s_{0}^{2}$ and we may have $s_{0}=0$ or $s_{0}=1$ . In the case that $s_{0}=1$ , then $s_{i}=0$ , $\forall i\geq 1$ , and $\tilde{\tau}(t)=t$ . Whereas, if $s_{0}=0$ , then $s_{1}=s_{1}^{2}$ and $s_{1}=0$ or $s_{1}=1$ . In the case that $s_{1}=1$ , then $s_{i}=0$ , $\forall i\geq 2$ , and $\tilde{\tau}(t)=t$ .

In the case that $s_{1}=0$ , then $s_{2}=s_{2}^{2}$ so $s_{2}=0$ or $s_{2}=1$ . Repeating the same reasoning and exploiting (11), we get in general that $\tilde{\tau}(t)=t^{k}$ must hold for a fixed $k\geq 1$ .

Let us consider $A(t)=\frac{1}{1-t}$ , then

[TABLE]

By definition, $\tilde{\tau}(\frac{1}{1-t})=\tilde{\tau}(\sum_{n=0}^{+\infty}t^{n})=\sum_{n=0}^{+\infty}(\tau(\mathbf{1}))_{n}t^{n}$ and $\tilde{\tau}(\frac{1}{1-t})=\sum_{n=0}^{+\infty}t^{kn}$ . Comparing the coefficients, we have

[TABLE]

Thus, by (10) and the definition of $\tau$ and $\tilde{\tau}$ , we have

[TABLE]

which is not a linear recurrent sequence. ∎

Conflict of interest

The authors assert that there are no conflicts of interest.

Bibliography24

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] M. Abrate, S. Barbero, U. Cerruti, N. Murru, Colored compositions, Invert operator and elegant compositions with the ”black tie” , Discrete Mathematics 335, 1–7, (2014).
2[2] S. Barbero, U. Cerruti, N. Murru, Some combinatorial properties of the Hurwitz series ring , Ricerche di Matematica 67, 491–-507, (2018).
3[3] S. Barbero, U. Cerruti, N. Murru, On the operations of sequences in rings and binomial type sequences , Ricerche di Matematica 67, 911–927, (2018).
4[4] A. Benhissi, Ideal structure of Hurwitz series rings , Contributions to Algebra and Geometry 48, 251-256, (2007).
5[5] A. Benhissi, F. Koja, Basic properties of Hurwitz series rings , Ricerche di Matematica 61, 255-273, (2012).
6[6] E. Cakcak, A remark on the minimal polynomial of the product of linear recurring sequences , Finite Fields and their Applications 4, 87–97, (1998).
7[7] U. Cerruti, F. Vaccarino, R–Algebras of linear recurrent sequences , J. Algebra 175(1), 332–338, (1995).
8[8] G. Everest, A. van der Poorten, I. Shparlinski, T. Ward, Recurrence sequences , Mathematical Surveys and Monographs 104, (2003).

TL;DR

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Abstract

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Some notes on the algebraic structure of linear recurrent sequences

Abstract.

1. Introduction

2. Preliminaries and notation

Remark 1**.**

Remark 2**.**

Lemma 3**.**

Proof.

Definition 4**.**

3. R-algebras of linear recurrent sequences

Theorem 5**.**

Proof.

Remark 6**.**

Proposition 7**.**

Proof.

Remark 8**.**

Theorem 9**.**

Proof.

Proposition 10**.**

Proof.

Remark 11**.**

4. On isomorphisms between R-algebras

Theorem 12**.**

Proof.

Theorem 13**.**

Proof.

Theorem 14**.**

Proof.

Conflict of interest

Remark 1.

Remark 2.

Lemma 3.

Definition 4.

Theorem 5.

Remark 6.

Proposition 7.

Remark 8.

Theorem 9.

Proposition 10.

Remark 11.

Theorem 12.

Theorem 13.

Theorem 14.