1 Introduction

Around the turn of the millennium, [1, 14] independently considered the Dirichlet series \(\sum _{n=1}^{\infty } F_{n}^{-s}\) for the Fibonacci sequence, establishing its meromorphic continuation to the complex plane. In addition, a problem in the American Mathematical Monthly posed this question for a recurrence of degree 2 of the form \(a \alpha ^{n} + b \alpha ^{-n}\), where \(a,b > 0\) and \(\alpha > 1\), corresponding to the quadratic polynomial \(x^{2} - (\alpha + \alpha ^{-1}) x + 1\). In [5] the same is done for a general quadratic recurrence.

In addition [14] gives explicit formulas for the residues, for the finite values at negative integers, and also some relations for values at positive integers. The latter question specifically has also been studied and generalized, for quadratic recurrences in [2, 4, 15], for the Tribonacci sequence in [7], and for a recurrence of arbitrary degree in [6].

Recently [20] consider a Hurwitz-type zeta function, namely arising from the meromorphic continuation of a series of the form \(\sum _{n=1}^{\infty } (a_{n} + x)^{-s}\) for a general Lucas quadratic sequence with real coefficients and the same authors in [19] consider a general real cubic recurrence, giving the Tribonacci sequence as an example.

Finally, we should mention [10, 11], where the authors consider an analogous multiple-Lucas zeta functions and twists of these by Dirichlet characters.

In this paper, we generalize most of these results, considering a general recurrence sequence \(a_{n}\) of arbitrary degree and the Dirichlet series \(\sum _{n=1}^{\infty } a_{n}^{-s}\). We will restrict ourselves to recurrences over the integers because we are specially interested in arithmetic properties of the finite values of the analytic continuation, for example their rationality at negative integers. Strictly speaking, this restriction is not necessary for many of our results, in particular for establishing the analytic continuation.

We begin in Sect. 2 recalling the basic results we need on recurrence sequences over the integers. In Sect. 3 we take up the study of the associated Dirichlet series and establish its basic properties. Proposition 3.4 gives conditions that ensure that the Dirichlet series associated to a recurrence is well-defined, setting the stage for further analysis.

Section 4 is the primary focus of the paper. The main result is Theorem 4.1, which proves the meromorphic continuation of the Dirichlet series associated to a recurrence satisfying the above-mentioned conditions, providing us with a definition of the zeta function associated to the recurrence. We also completely determine its pole set, and in Corollary 4.7 we compute the residues.

Section 5 reviews some special cases, namely, quadratic and cubic recurrences, including Fibonacci and Tribonacci recurrences, and as an example of general degree, N-bonacci sequences.

Finally, in Sect. 6, using standard arguments from Galois theory, we prove the second main result (Theorem 6.1), which establishes the rationality of the values of a recurrence zeta function at negative integers which are not poles.

2 General Remarks About Recurrence Sequences

Before going into the study of the Dirichlet series associated to a linear recurrence sequence, let us recall a few known facts about such sequences that will be useful in what follows.

Let us formalize what we mean by “linear recurrence sequence”. First, for a commutative ring R with identity \(1\in R\), let \(R^{\infty }\) be the set of sequences of elements of R, that is, of mappings \({\mathbb {N}}\rightarrow R\). Note that this is an R-module. We can define in \(R^{\infty }\) the R-linear operator \(\nabla \) as

$$\begin{aligned} \begin{array}{rccl} \nabla : &{} R^{\infty } &{} \longrightarrow &{} R^{\infty } \\ {} &{} \{a_n\}_{n\in {\mathbb {N}}} &{} \longmapsto &{} \{a_{n+1}\}_{n\in {\mathbb {N}}}\end{array}. \end{aligned}$$

For brevity, we write \(\nabla a_n=a_{n+1}\). Since this operator is R-linear, it makes sense to consider, for a polynomial \(Q(x)\in R[x]\), the operator \(Q(\nabla )\), and this way we can give \(R^{\infty }\) a R[x]-module structure, where

$$\begin{aligned} Q(x)\cdot \{a_n\}=Q(\nabla )\{a_n\}. \end{aligned}$$

This is valid for any ring, but let us specialize now to the case \(R={\mathbb {Z}}\), which will be the case in everything that follows.

Definition 2.1

We say that \(\{a_n\}\in {\mathbb {Z}}^{\infty }\) is a linear recurrence sequence (LRS) if there is a monic polynomial \(Q(x)\in {\mathbb {Z}}[x]\) such that \(Q(\nabla )a_n=0\).

For a LRS \(\{a_n\}\) over \({\mathbb {Z}}\), we can define its annihilator ideal by

$$\begin{aligned} A_{{\mathbb {Z}}}(\{a_n\})=\{Q(x)\in {\mathbb {Z}}[x]\,\vert \, Q(\nabla )a_n=0\}\subseteq {\mathbb {Z}}[x]. \end{aligned}$$

If \(\{a_n\}\ne 0\), this is a proper ideal in \({\mathbb {Z}}[x]\). Thanks to Gauss’ lemma, we have the following result:

Proposition 2.2

Let \(\{a_n\}\) be a LRS over \({\mathbb {Z}}\). There is a unique monic polynomial \(P(x)\in {\mathbb {Z}}[x]\) such that \(A_{{\mathbb {Z}}}(\{a_n\})=(P)\).

Proof

Let \(Q(x)\in {\mathbb {Z}}[x]\) be a monic polynomial in \(A_{{\mathbb {Z}}}(\{a_n\})\), which exists because \(\{a_n\}\) is a LRS over \({\mathbb {Z}}\). Since \(A_{{\mathbb {Z}}}(\{a_n\})\subseteq A_{{\mathbb {Q}}}(\{a_n\})\) and \(A_{{\mathbb {Q}}}(\{a_n\})=(P)\) for some unique monic \(P(x)\in {\mathbb {Q}}[x]\) (because \({\mathbb {Q}}[x]\) is a principal ideal domain), P(x) divides Q(x) and it is the product of some of the irreducible factors of Q(x) over \({\mathbb {Q}}\). However, due to Gauss’ lemma, these factors are in \({\mathbb {Z}}\), and therefore \(P(x)\in {\mathbb {Z}}[x]\). It is now straightforward that \(A_{{\mathbb {Z}}}(\{a_n\})=(P)\). \(\square \)

Thanks to Proposition 2.2, the following definition makes sense:

Definition 2.3

Let \(\{a_n\}\) be a LRS over \({\mathbb {Z}}\). The unique monic polynomial \(P(x)\in {\mathbb {Z}}[x]\) such that \(A_{{\mathbb {Z}}}(\{a_n\})=(P)\) is called the minimal polynomial of the LRS \(\{a_n\}\). It is minimal in the sense that it is the monic polynomial of least degree in \(A_{{\mathbb {Z}}}(\{a_n\})\).

Remark 2.4

If \(\{a_n\}\) is annihilated by an irreducible monic polynomial, it is clear from the proof of Proposition 2.2 that it must be its minimal polynomial. However, unlike the case of algebraic integers, minimal polynomials of LRS need not be irreducible: take, for example, \(a_n=2^n+3^n\). It is in a LRS because \(a_{n+2}=5a_{n+1}-6a_n\) for all \(n\in {\mathbb {N}}\). This means that the polynomial \(P(x)=x^2-5x+6=(x-2)(x-3)\) annihilates it, but it is easy to check that none of its factors do, so P(x) is the minimal polynomial.

The last result that we need with respect to LRS is what we may call a “Binet-like” formula, due to its similarity with the Binet formula for the Fibonacci sequence, which this generalises. It can be found in [3], §1.1.6 for any base field with characteristic 0, we state it for \({\mathbb {Q}}\):

Proposition 2.5

Let \(\{a_n\}\) be a LRS over \({\mathbb {Z}}\) with minimal polynomial \(P(x)\in {\mathbb {Z}}[x]\). Let K be the splitting field of P(x) over \({\mathbb {Q}}\) and \(\alpha _1,\ldots ,\alpha _r\) the roots of P(x) in K, each with multiplicity \(m_i\). There are polynomials \(\lambda _1(x),\ldots ,\lambda _r(x)\in K[x]\) (in fact, \(\lambda _i(x)\in {\mathbb {Q}}(\alpha _i)[x]\)), with \({\text {deg}}\lambda _i\le m_i-1\), such that

$$\begin{aligned} a_n=\lambda _1(n)\alpha _1^n+\cdots +\lambda _r(n)\alpha _r^n \quad \forall n\in {\mathbb {N}}. \end{aligned}$$

In particular, if the minimal polynomial P(x) of \(\{a_n\}\) is separable with nonzero constant term, then \(a_n\) can be expressed as a linear combination of the powers of the roots of P(x). In this case, the coefficients \(\lambda _i(x)=\lambda _i\in K\) can be obtained easily through a system of linear equations:

$$\begin{aligned} \left( \begin{array}{c} \lambda _1 \\ \vdots \\ \lambda _r\end{array}\right) =\left( \begin{array}{ccc} \alpha _1 &{} \ldots &{} \alpha _r \\ \vdots &{} \ddots &{} \vdots \\ \alpha _1^r &{} \ldots &{} \alpha _r^r\end{array}\right) ^{-1}\left( \begin{array}{c} a_1 \\ \vdots \\ a_r\end{array}\right) . \end{aligned}$$
(2.6)

3 Dirichlet Series Defined by a LRS

A general Dirichlet series is an expression of the form

$$\begin{aligned} \sum _{n=1}^{\infty } b_ne^{-\lambda _ns}, \end{aligned}$$
(3.1)

where s is a complex variable, \(\{b_n\}\) a sequence of complex numbers and \(\{\lambda _n\}\) a strictly increasing sequence of nonnegative real numbers such that \(\lambda _n\rightarrow \infty \). A classical Dirichlet series is a Dirichlet series for which \(\lambda _n=\log n\).

Dirichlet series have been studied in depth and for a long time. Let us give a summary of some well-known results about them here:

There is a real number (possibly \(-\infty \) or \(+\infty \)) \(\sigma _c\) such that the series (3.1) converges for \(\sigma =\Re (s)>\sigma _c\) and diverges for \(\sigma <\sigma _c\): it is called the convergence abscissa of the series. It can be computed by:

$$\begin{aligned} \sigma _c={\left\{ \begin{array}{ll} \lim \sup \frac{\log |b_1+\cdots +b_n|}{\lambda _n}\qquad &{}\text {if }\sum b_k\text { diverges}, \\ \lim \sup \frac{\log |b_{n+1}+b_{n+2}+\cdots |}{\lambda _n}\qquad &{}\text {if }\sum b_k\text { converges}.\end{array}\right. } \end{aligned}$$
(3.2)

There is also a real number \(\sigma _a\) (again, possible \(\pm \infty \)) such that the series (3.1) converges absolutely for \(\sigma >\sigma _a\) and does not for \(\sigma <\sigma _a\). Applying the formula (3.2) to the series \(\sum |b_n e^{-\lambda _n s}|\), we get that

$$\begin{aligned} \sigma _a={\left\{ \begin{array}{ll} \lim \sup \frac{\log (|b_1|+\cdots +|b_n|)}{\lambda _n}\qquad &{}\text {if }\sum |b_k|\text { diverges}, \\ \lim \sup \frac{\log (|b_{n+1}|+\cdots )}{\lambda _n}\qquad &{}\text {if }\sum |b_k|\text { converges}.\end{array}\right. } \end{aligned}$$
(3.3)

It is always true that

$$\begin{aligned} 0\le \sigma _a-\sigma _c\le \lim \sup \frac{\log n}{\lambda _n}. \end{aligned}$$

For the proof of all these statements, the reader can check Sect. 207 in [9].

In the case of classical Dirichlet series, \(0\le \sigma _a-\sigma _c\le 1\) (for example, the alternating Dirichlet series \(\eta (s)=\sum (-1)^{n+1}n^{-s}\) has \(\sigma _c=0\) and \(\sigma _a=1\)).

We want to study Dirichlet series of the type

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{1}{a_n^s}=\sum _{n=1}^{\infty } e^{-s\log a_n }, \end{aligned}$$

where \(\{a_n\}\) is a LRS over \({\mathbb {Z}}\). Note that, if this is to be coherent with our definition of a Dirichlet series, we need the sequence \(\{a_n\}\) to be a strictly increasing sequence of positive integers. However, we can take a finite number of terms out of the sum (3.1) without changing its nature too much (since a finite sum of exponential functions is a holomorphic function), we will allow for sequences that are eventually strictly increasing, that is, sequences for which there is some \(n_0\in {\mathbb {N}}\) such that \(\{a_{n+n_0}\}_{n\in {\mathbb {N}}}\) is strictly increasing (eventual positivity follows from this). A sufficient condition for this to happen is the following:

Proposition 3.4

Let \(\{a_n\}\) be a LRS over \({\mathbb {Z}}\) with minimal polynomial P(x). Let \(\alpha _1,\ldots ,\alpha _r\) be the roots of P(x), each with multiplicity \(m_i\), and put

$$\begin{aligned} a_n=\lambda _1(n)\alpha _1^n+\cdots +\lambda _r(n)\alpha _r^n, \end{aligned}$$

as in Proposition 2.5. Assume there is one root \(\alpha _{i_0}\), with \(\lambda _{i_0}(n)\) not identically 0, verifying \(|\alpha _{i_0}|>|\alpha _i|\) for every \(i\ne i_0\) such that \(\lambda _{i}(n)\) is not identically 0 and \(|\alpha _{i_0}|>1\). Then, there are two exhaustive and mutually exclusive possibilities:

  1. 1.

    Either \(\{a_n\}\) or \(\{-a_n\}\) is eventually strictly increasing.

  2. 2.

    \(\{a_n\}\) has an infinite number of sign changes as n grows to \(\infty \).

Proof

Assume, without loss of generality, that \(i_0=1\).

First, note that the hypothesis implies that \(\alpha _1\in \mathbb {R}\): \(A_{{\mathbb {C}}}(\{a_n\})\) contains the polynomial \(Q(x)=(x-\alpha _{i_1})...(x-\alpha _{i_t})\), where \(\{i_1,\ldots ,i_t\}\) is the set of indices such that \(\lambda _i(n)\) is not identically 0. If it were not \(Q(x)\in {\mathbb {Z}}[x]\), then \(\{a_n\}\) could not be a sequence of integers. This means that, if \(\lambda _{j_1}(n)\) is not identically 0 and \(\alpha _{j_2}\) is the conjugate root of \(\alpha _{j_1}\), then \(\lambda _{j_2}(n)\) is also not identically 0. Since, by hypothesis, there is no other root with modulus as large as \(|\alpha _1|\), \(\alpha _1\) does not have a complex conjugate and it is real.

Now, since \(\lambda _1(x)\) is a polynomial (and it must have real coefficients since \(\alpha _1\in {\mathbb {R}}\)), there is some \(n_0\gg 0\) such that \(\lambda _1(n)\) has constant sign, equal to the sign of the leading coefficient of \(\lambda _1(x)\), for every \(n\ge n_0\). In particular, it is nonzero, so for \(n\ge n_0\) we can put

$$\begin{aligned} a_n=\lambda _1(n)\alpha _1^n\left( 1+\frac{\lambda _2(n)}{\lambda _1(n)} \left( \frac{\alpha _2}{\alpha _1}\right) ^n+\cdots +\frac{\lambda _r(n)}{\lambda _1(n)}\left( \frac{\alpha _r}{\alpha _1}\right) ^n\right) . \end{aligned}$$

For every \(i=2,\ldots ,r\), it is clear that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\lambda _i(n)}{\lambda _1(n)}\left( \frac{\alpha _r}{\alpha _1}\right) ^n=0, \end{aligned}$$

so there is some \(n_1>n_0\) such that, for \(n\ge n_1\), \(a_n\) has the same sign as \(\lambda _1(n)\alpha _1^n\). If \(\alpha _1>1\), this sign is that of the leading coefficient of \(\lambda _1(x)\), and the fact that \(a_n\) is eventually strictly increasing is obvious since \(\alpha _1>1\). If \(\alpha _1<-1\), this sign changes with the parity of n. \(\square \)

Remark 3.5

The condition used in Proposition 3.4 is, while sufficient, not always necessary for a LRS to be eventually strictly increasing sequence. For example, the polynomial \(x^2-2\) has 2 roots with the same absolute value, and for any initial terms \(a_1,a_2\) such that \(0<a_1<a_2<2a_1\), the LRS \(\{a_n\}\) that this defines is strictly increasing. However, the existence of one dominant root \(>1\) will be a necessary hypothesis in the proof of Theorem 4.1, which is our main result, so there is no need to study here other kinds of eventually strictly increasing LRS.

From now on we will deal with LRS \(\{a_n\}\) whose minimal polynomial is irreducible over \({\mathbb {Q}}\) (from now on, we will refer only to irreducibility over \({\mathbb {Q}}\)). From the discussion in the proof of Proposition 3.4 it is clear that, in this case, all the coefficients \(\lambda _i(n)\) are not identically 0. Furthermore, since irreducible polynomials are also separable, these coefficients are constant. Then, if there is a positive dominant root (a root whose modulus is greater than that of the other roots), \(\{a_n\}\) is automatically eventually strictly increasing or eventually strictly decreasing, because this root (except on the case \(P(x)=(x-1)\), which gives constant LRS) is also automatically greater than 1.

Let \(\{a_n\}\) be a LRS over \({\mathbb {Z}}\) such that its minimal polynomial P(x) is irreducible and has a positive dominant root (\(\alpha _1\)) and its coefficient in the Binet formula for \(a_n\) is positive. Since there is some \(n_0\gg 0\) such that \(\{a_{n+n_0}\}_{n\in {\mathbb {N}}}\) is strictly increasing and positive, we will assume, without loss of generality in what follows, that \(\{a_n\}\) is strictly increasing and positive. We can therefore define the Dirichlet series associated to the sequence \(\{a_n\}\) as

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{1}{a_n^s}=\sum _{n=1}^{\infty } e^{-s\log a_n } \end{aligned}$$

Using both formula (3.2) and Binet’s formula 2.5 we can deduce easily that its abscissa of convergence is

$$\begin{aligned} \sigma _c=\lim \sup \frac{\log n}{\log a_n}=\lim \frac{\log n}{\log \lambda _1+n\log \alpha _1}=0, \end{aligned}$$

and since the coefficients \(b_n=1\) of the series are all positive, the same goes for its abscissa of absolute convergence. In the next section we will show that this series, that defines a holomorphic function in the half-plane \(\sigma >0\), has a meromorphic continuation to the whole s-plane, and we will find its poles and residues.

4 Meromorphic Continuation of the Dirichlet Series

Let, as before, \(\{a_n\}\) be a LRS of integers with minimal polynomial \(P(x)\in {\mathbb {Z}}[x]\) irreducible and with a positive dominant root. Let \(\alpha _1,\ldots ,\alpha _r\) be the roots of P(x) (ordered by decreasing modulus, so that \(\alpha _1\) is the dominant root) and \(\lambda _1,\ldots ,\lambda _r\) nonzero numbers in the splitting field of P(x) such that

$$\begin{aligned} a_n=\lambda _1\alpha _1^n+\cdots +\lambda _r\alpha _r^n. \end{aligned}$$

Assume once again, without loss of generality, that \(\{a_n\}\) is strictly increasing and positive (that is, \(\lambda _1>0\)). Then, as per the discussion at the end of the previous section, the Dirichlet series

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{1}{a_n^s} \end{aligned}$$

converges and does so absolutely in the half-plane \(\sigma =\Re (s)>0\), so it defines a holomorphic function \(\varphi (s)\) there.

Theorem 4.1

With the previous hypotheses, the holomorphic function \(\varphi (s)\) defined by the Dirichlet series associated to \(\{a_n\}\) can be continued analytically to a meromorphic function on \({\mathbb {C}}\) (that we will still call \(\varphi (s)\)) whose only singularities are simple points at the poles

$$\begin{aligned} s_{n,k_1,\ldots ,k_{r-1}}=\frac{\log |\alpha _1^{-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}|}{\log \alpha _1} +i\frac{\arg (\alpha _1^{-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}})+2\pi n}{\log \alpha _1}, \end{aligned}$$
(4.2)

where the parameters \(n,k_1,\ldots ,k_{r-1}\) are integers satisfying

$$\begin{aligned} n\in {\mathbb {Z}}, \qquad k_1\ge 0, \qquad 0\le k_i\le k_{i-1}\quad i=2,\ldots ,r-1. \end{aligned}$$

Proof

Using Binet’s formula for \(a_n\) we can expand \(a_n^{-s}\) using the binomial series to obtain

$$\begin{aligned} a_n^{-s}= & {} \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{k_1}...\sum _{k_{r-1}=0}^{k_{r-2}}\left( {\begin{array}{c}-s\\ k_1\end{array}}\right) \left( {\begin{array}{c}k_1\\ k_2\end{array}}\right) ... \\{} & {} \left( {\begin{array}{c}k_{r-2}\\ k_{r-1}\end{array}}\right) (\lambda _1\alpha _1^n)^{-s-k_1}(\lambda _2\alpha _2^n)^{k_1-k_2}...(\lambda _r\alpha _r^n)^{k_{r-1}}. \end{aligned}$$

Note that this is technically only valid for the \(a_n\) in which

$$\begin{aligned} \left| \frac{\lambda _2\alpha _2^n+\cdots +\lambda _r\alpha _r^n}{\lambda _1\alpha _1^n}\right| <1, \end{aligned}$$

but, due to the assumption that \(\alpha _1\) is a dominant root, this holds for all but a finite number of n, and we can assume without loss of generality for the purpose of analytic continuation that it is the case for every \(n\in {\mathbb {N}}\), since the terms of the Dirichlet series where it does not hold add up to an entire function.

In order to try to simplify our notation a bit we set

$$\begin{aligned} k_{\le j}=\{k_1,\ldots ,k_j\}, \end{aligned}$$

so that

$$\begin{aligned} \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{k_1}...\sum _{k_{r-1}=0}^{k_{r-2}}=\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}}\quad \text { and }\quad \left( {\begin{array}{c}-s\\ k_1\end{array}}\right) \left( {\begin{array}{c}k_1\\ k_2\end{array}}\right) ...\left( {\begin{array}{c}k_{r-2}\\ k_{r-1}\end{array}}\right) =\left( {\begin{array}{c}-s,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) . \end{aligned}$$

This way, the Dirichlet series \(\sum a_n^{-s}\) can be written as

$$\begin{aligned} \sum _{n=1}^{\infty }\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}}\left( {\begin{array}{c}-s,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) (\lambda _1\alpha _1^n)^{-s-k_1}(\lambda _2\alpha _2^n)^{k_1-k_2}...(\lambda _r\alpha _r^n)^{k_{r-1}}, \end{aligned}$$

or

$$\begin{aligned} \sum _{n=1}^{\infty }\sum _{k_{\le r-1}=0}^{\infty ,k_{\le r-2}}\Lambda _{k_{\le r-1}}(s)\left( \alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}\right) ^n, \end{aligned}$$
(4.3)

where we write

$$\begin{aligned} \Lambda _{k_{\le r-1}}(s)=\left( {\begin{array}{c}-s,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) \lambda _1^{-s-k_1}\lambda _2^{k_1-k_2}...\lambda _r^{k_{r-1}}, \end{aligned}$$

to shorten the expression further. Since these functions have no poles, they do not change the reasoning that follows.

Now let us see that the series (4.3) is absolutely convergent as a double series for \(\sigma =\Re (s)>0\). For the functions \(\Lambda _{k_{\le r-1}}(s)\) we have the estimate, as in [14],

$$\begin{aligned} \left| \Lambda _{k_{\le r-1}}(s)\right| \le (-1)^{k_1}\left( {\begin{array}{c}-|s|,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) \lambda _1^{-\sigma -k_1}|\lambda _2|^{k_1-k_2}...|\lambda _r|^{k_{r-1}}. \end{aligned}$$

Using this, we get that

$$\begin{aligned}{} & {} \sum _{n=1}^{\infty }\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}}\left| \Lambda _{k_{\le r-1}}(s)\left( \alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}\right) ^n\right| \le \nonumber \\{} & {} \quad \le \sum _{n=1}^{\infty }\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}}(-1)^{k_1}\left( {\begin{array}{c}-|s|,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) (\lambda _1\alpha _1^n)^{-\sigma -k_1}...|\lambda _r\alpha _r^n|^{k_{r-1}}= \nonumber \\{} & {} \quad =\lambda _1^{-\sigma }\sum _{n=1}^{\infty }\alpha _1^{-n\sigma }\left( 1-\frac{|\lambda _2\alpha _2^n|+\cdots +|\lambda _r\alpha _r^n|}{\lambda _1\alpha _1^n}\right) ^{-|s|}\le \nonumber \\{} & {} \quad \le \lambda _1^{-\sigma }\left( 1-\frac{|\lambda _2\alpha _2|+\cdots +|\lambda _r\alpha _r|}{\lambda _1\alpha _1}\right) ^{-|s|}\sum _{n=1}^{\infty }\alpha _1^{-n\sigma }<+\infty . \end{aligned}$$
(4.4)

Note that the step between lines 3 and 4 assumes that the sequence of real numbers

$$\begin{aligned} \frac{|\lambda _2\alpha _2^n|+\cdots +|\lambda _r\alpha _r^n|}{\lambda _1\alpha _1^n} \end{aligned}$$

is strictly decreasing and always less than 1. Since this is eventually true (that is, true for \(n\ge n_0\) for some \(n_0\in {\mathbb {N}}\)) and, again, a finite number of the terms of the series only change the total by an entire function, we can assume \(n_0=0\) without loss of generality.

From the reasoning (4.4), the double series (4.3) is absolutely convergent for \(\sigma >0\), so we can change the order of summation and, since

$$\begin{aligned} \left| \alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}\right| =\alpha _1^{-\sigma -k_1}|\alpha _2|^{k_1-k_2}...|\alpha _r|^{k_{r-1}}<1 \end{aligned}$$

for \(\sigma >0\), \(k_1\ge 0\), the double series becomes

$$\begin{aligned} \varphi (s)=\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}} \Lambda _{k_{\le r-1}}(s)\frac{\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}{1-\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}. \end{aligned}$$
(4.5)

It remains to see that \(\varphi (s)\) is a meromorphic function. Once this is done, the statement about its poles is trivial. To do this, we claim that the series (4.5) converges uniformly on every compact set of the s-plane that does not contain any of the points \(s_{n,k\le r-1}\) defined as in (4.2).

Let us set

$$\begin{aligned} f_{k_{\le r-1}}(s)=\Lambda _{k_{\le r-1}}(s)\frac{\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}{1-\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}. \end{aligned}$$

For any \(s\in {\mathbb {C}}\) we have, for \(k_1\gg 0\),

$$\begin{aligned} \begin{aligned} \left| \alpha _1^{s+k_1}\alpha _2^{-k_1+k_2}...\alpha _r^{-k_{r-1}}-1\right|&\ge \alpha _1^{\sigma +k_1}|\alpha _2|^{-k_1+k_2}...|\alpha _r|^{-k_{r-1}}-1> \\&>\alpha _1^{\sigma +k_1-1}|\alpha _2|^{-k_1+k_2}...|\alpha _r|^{-k_{r-1}}, \end{aligned} \end{aligned}$$

so there is some \(k_0\gg 0\) such that

$$\begin{aligned} \begin{aligned}&\sum _{k_1=k_0}^{\infty }\sum _{k_2=0}^{k_1}... \sum _{k_{r-1}=0}^{k_{r-2}}\left| f_{k_{\le r-1}}(s)\right| \le \\ {}&\quad \le \lambda _1^{-\sigma }\alpha _1^{-\sigma +1}\sum _{k_{\le r-1}=0}^{\infty , k_{\le r-2}} (-1)^{k_1}\left( {\begin{array}{c}-|s|,k_{\le r-2}\\ k_{\le r-1}\end{array}}\right) (\lambda _1\alpha _1)^{-k_1}...|\lambda _r\alpha _r|^{k_{r-1}} \\&\quad =\lambda _1^{-\sigma }\alpha _1^{-\sigma +1}\left( 1-\frac{|\lambda _2\alpha _2|+\cdots +|\lambda _r\alpha _r|}{\lambda _1\alpha _1}\right) ^{-|s|}<+\infty . \end{aligned} \end{aligned}$$

This bound is uniform when s varies in a compact set, so we get that \(\varphi (s)\), as defined by (4.5), is a meromorphic function of s with simple poles at the points stated in (4.2). \(\square \)

Remark 4.6

Note that we have not said that all the points \(s_{n,k_{\le r-1}}\) are distinct. It could happen that, for two different r-tuples \(n,k_1,\ldots ,k_{r-1}\) and \(n',k_1',\ldots ,k_{r-1}'\), \(s_{n,k_{\le r-1}}=s_{n',k'_{\le r-1}}\). However, it is not hard to check that any pole \(s_0\) of \(\varphi (s)\) can only be reached by a finite number of r-tuples, so this observation does not change the truth of the statements in Theorem 4.1.

Though in general we do not know the answer to this question (i.e., Are the points \(s_{n,k_{\le r-1}}\) distinct for different r-tuples?), it is possible to answer it in the affirmative in particular cases, as we will see later in some examples.

Corollary 4.7

Following the notation of Theorem 4.1, if \(s_0\in {\mathbb {C}}\) is a pole of \(\varphi (s)\) and \(\varvec{\tau }(s_0)\) denotes the set of r-tuples \((n,k_1,\ldots ,k_{r-1})\) such that \(s_{n,k_{\le r-1}}=s_0\), the residue of \(\varphi (s)\) at the pole \(s=s_0\) is

$$\begin{aligned} \sum _{(n,k_{\le r-1})\in \varvec{\tau }(s_0)}\frac{1}{\log \alpha _1}\lambda _1^{-s_0-k_1} \lambda _2^{k_1-k_2}..\lambda _r^{k_{r-1}} \left( {\begin{array}{c}-s_0\\ k_1\end{array}}\right) \left( {\begin{array}{c}k_1\\ k_2\end{array}}\right) ...\left( {\begin{array}{c}k_{r-2}\\ k_{r-1}\end{array}}\right) . \end{aligned}$$
(4.8)

Proof

First of all, note that, as we stated in Remark 4.6, if \(s_0\in {\mathbb {C}}\) is a pole of \(\varphi (s)\) then \(\varvec{\tau }(s_0)\) is nonempty and finite. Therefore, there is no obstacle to the calculation

$$\begin{aligned} \begin{aligned} {\text {Res}}_{s=s_0}\varphi (s)&=\lim _{s\rightarrow s_0}(s-s_0)\varphi (s)= \\ {}&=\sum _{\varvec{\tau }(s_0)}\lim _{s\rightarrow s_0}(s-s_0)f_{k_{\le r-1}}(s_0)=\\&=\sum _{\varvec{\tau }(s_0)}\lim _{s\rightarrow s_0}\frac{(s-s_0)\Lambda _{k_{\le r-1}}(s)\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}{1-\alpha _1^{-s-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}= \\ {}&=\sum _{\varvec{\tau }(s_0)}\frac{\Lambda _{k_{\le r-1}}(s_0)}{\log \alpha _1}, \end{aligned} \end{aligned}$$

and using the explicit definition of \(\Lambda _{k_{\le r-1}}(s)\) we get (4.8). \(\square \)

5 Some Particular Series

Let us now study some concrete examples of Dirichlet series associated to a LRS.

Example 1

(Quadratic recurrence) The first case that we focus on, which is both the simplest and the most studied, is the quadratic recurrence relation. That is, the LRS \(\{a_n\}\) that defines our Dirichlet series has an irreducible minimal polynomial \(P(x)\in {\mathbb {Z}}[x]\) of degree 2 with a dominant root.

The most famous of this sequence is the Fibonacci sequence \(\{F_n\}\), with initial terms \(F_1=F_2=1\) and minimal polynomial \(x^2-x-1\). Its associated Dirichlet series was studied in depth by Navas in [14] and by Egami in [1] (although note that the list of poles given there also contains some points which are in fact removable singularities). Its companion, the Lucas sequence, and their generalizations to LRS of order 2 also define Dirichlet series that were studied by Kamano in [5]. There are also studies about the values of these series on positive integers, like [4, 15]. The meromorphic continuation for a general sequence of order 2 with positive coefficients is studied also in [17]. More recently [20] generalizes these series to a Hurwitz-type zeta function associated to a Lucas sequence.

If the minimal polynomial \(P(x)=x^2+p_1x+p_0\in {\mathbb {Z}}[x]\) is irreducible and has one dominant root \(\alpha >1\), the other root will be \(\widetilde{\alpha }=p_0\alpha ^{-1}\), or \({{\text {N}}}(\alpha )\alpha ^{-1}\), if \({{\text {N}}}\) denotes the absolute norm in \({\mathbb {Q}}(\alpha )\). Then, our formula (4.2) for the poles of its associated Dirichlet series yields

$$\begin{aligned} s_{n,k}=-k\left( 2-\frac{\log |{{\text {N}}}(\alpha )|}{\log \alpha }\right) +i\frac{k\arg (\widetilde{\alpha })+2\pi n}{\log \alpha _1}, \qquad n\in {\mathbb {Z}}, \quad k\ge 0. \end{aligned}$$

Note that \(\arg (\widetilde{\alpha })\) is either 0 or \(\pi \), so in any case the imaginary part of \(s_{n,k}\) is an integer multiple of \(\pi \) divided by \(\log \alpha _1\).

It is clear that in this case, all of these points are distinct (remember that, as per Remark 4.6, this is not proved in the general case), and they form a semi-lattice in the half-plane \(\Re (s)\le 0\). If, for example, \({{\text {N}}}(\alpha )=\pm 1\), that is, \(\alpha \) is a unit (as is the case, for example, in the Fibonacci sequence), these poles will be arranged in vertical lines with abscissa \(-2k\), \(k\ge 0\), and the other direction of the semi-lattice will be horizontal or oblique depending on whether \({{\text {N}}}(\alpha )=+1\) or \({{\text {N}}}(\alpha )=-1\).

Navas proved, in [14], that, in the case of the Fibonacci Dirichlet series, the values at the negative integers that are not poles are rational numbers, and derived formulas for them that depend on the Fibonacci and Lucas numbers. The first part of this result is proved in general in the following section §6.

Example 2

(Cubic recurrence) The second case, which has some more variety and is much less known, is the cubic recurrence relation. That is, the LRS \(\{a_n\}\) that defines our Dirichlet series has an irreducible minimal polynomial \(P(x)\in {\mathbb {Z}}[x]\) of degree 3 with a dominant root. An example would be the Tribonacci sequence, defined by the recurrence relation \(T_{n+3}=T_{n+2}+T_{n+1}+T_n\) with initial values \(T_1=T_2=1\), \(T_3=2\) (a displacement of sequence A000073 in [18] or [8]). The values at positive integers of the corresponding Dirichlet series were studied in [7], while the function defined by the series has been recently examined in [19], considering arbitrary real coefficients.

In general, for an irreducible cubic polynomial \(P(x)\in {\mathbb {Z}}[x]\) with one dominant root \(>1\), we have two possibilities: either the other two roots are complex conjugate, or they are both real (this last possibility, characterized by the discriminant of the polynomial being positive, is the casus irreducibilis).

In the case were there are two complex conjugate roots, we have one real root \(\alpha _1>1\) and two complex roots \(\alpha _2\), \(\alpha _3=\overline{\alpha _2}\). We have, ordering the roots appropriately \(\alpha _2=\rho e^{i\theta }\), with \(0<\rho <\alpha _1\) and \(0<\theta <\pi \), and \(\alpha _3=\rho e^{-i\theta }\). Once again, we can obtain \({{\text {N}}}(\alpha _1)=\alpha _1\rho ^2\in {\mathbb {N}}\), and the poles of the continuation of the Dirichlet series are the points

$$\begin{aligned} s_{n,k,l}=-\frac{k}{2}\left( 3-\frac{\log {{\text {N}}}(\alpha _1)}{\log \alpha _1}\right) +i\frac{(k-2l)\theta +2\pi n}{\log \alpha _1}, \quad n\in {\mathbb {Z}}, \quad 0\le l\le k. \end{aligned}$$

These poles, unlike in the case of quadratic recurrences, do not form a semi-lattice: they are still grouped by vertical lines (if, for example, \(\alpha _1\) is a unit, they are vertical lines of abscissa \(-\frac{3}{2}k\), \(k\ge 0\)), but in each of these lines the poles are increasingly “blurry” when we get away from the real axis.

Since the real part of these poles depends only on one parameter, whether or not all of these points are distinct depends solely on whether, for fixed k, \((k-2l)\theta +2\pi n\) can take repeated values when \(0\le l\le k\) and \(n\in {\mathbb {Z}}\). This is equivalent to asking whether or not \(\theta \) is a rational multiple of \(\pi \), which is equivalent to asking whether \(e^{i\theta }=\frac{\alpha _2}{|\alpha _2|}\) is a root of unity.

This question has a negative answer if \(\alpha _{1}\) is a cubic Pisot number with nonreal conjugates. This can be straightforwardly seen from the following result of [12]:

Theorem 5.1

Let \(\alpha \) be a Pisot number with conjugates \(\alpha = \alpha _{1}, \alpha _{2}, \ldots , \alpha _{d}\). Then the relation \(\alpha _{1}^{n_{1}} \alpha _{2}^{n_{2}} \dots \alpha _{d}^{n_{d}} = 1\) implies \(n_{1} = n_{2} = \cdots = n_{d}\).

Indeed, since \((\alpha _{2}/|\alpha _{2}|)^{n} = \alpha _{1}^{0} \alpha _{2}^{n} \alpha _{3}^{-n}\), if \((\alpha _{2}/|\alpha _{2}|)^{n} = 1\), the theorem implies \(n = 0\).

We do not know, in general, if this is the case for any cubic recurrence sequence with two complex roots, although there are examples where this happens. For instance, the polynomial \(x^{3} + x^{2} - 3 x - 5\) has a dominant root \(\alpha _{1}\) and the other two roots \(\alpha _{2}\), \(\alpha _{3}\) are complex conjugates lying ouside the unit disk. The quotient \(\alpha _{2}/|\alpha _{2}|\) is not a root of unity because its minimal polynomial is \(25 x^{12} + 60 x^{10} + 43 x^{8} + 12 x^{6} + 43 x^{4} + 60 x^{2} + 25\) and can be checked to have some roots of modulus different from 1.

For the casus irreducibilis, that is, the case where \(\alpha _1>1\) and \(\alpha _2,\alpha _3\in {\mathbb {R}}\), the structure of the poles is a bit different. Formula (4.2) gives us the points

$$\begin{aligned} s_{n,k,l}=\frac{-1}{\log \alpha _1}\left( k\log \left| \frac{\alpha _1}{\alpha _2}\right| +l\log \left| \frac{\alpha _2}{\alpha _3}\right| \right) +i\frac{k\arg (\alpha _2)+l\arg (\alpha _2^{-1}\alpha _3)+2\pi n}{\log \alpha _1}, \end{aligned}$$

where once again \(n\in {\mathbb {Z}}\), \(0\le l\le k\).

Since \(\alpha _2,\alpha _3\in {\mathbb {R}}\), the imaginary part is always a multiple of \(\frac{\pi }{\log \alpha _1}\), so these points are grouped into horizontal half-lines in the half-plane \(\Re (s)\le 0\), where in each horizontal line the points get “blurrier” when we travel away from the imaginary axis.

In this case Mignotte’s theorem also implies that if \(\alpha _{1}\) is a Pisot number, then the points \(s_{n,k,l}\) are distinct for different triples (n, k, l). Indeed, if \(s_{n,k,l} = s_{n',k',l'}\), comparing the real parts we obtain a relation \(\bigl |\alpha _{1}^{k-k'} \alpha _{2}^{(l - l') - (k - k')} \alpha _{3}^{-(l - l')}\bigr | = 1\). Since these are assumed real, squaring this relation and applying Mignotte’s theorem, we conclude that \(k - k' = (l - l') - (k - k') = -(l - l')\), from which one deduces that \(k = k'\) and \(l = l'\). Looking at the imaginary parts we conclude that \(n = n'\) also.

Example 3

(N-bonacci sequences) Let us speak shortly of one last example of these kind of Dirichlet series, this time associated to a sequence \(\{a_n\}\) satisfying the N-bonacci recurrence relation, i.e., whose minimal polynomial is \(P_{{\text {N}}}(x)=x^N-x^{N-1}-\cdots -x-1\). These polynomials are all irreducible, have one real root \(\phi _N\) in the interval (1, 2) and all their other roots lie in the open unit disk (see [13]). In fact, all other roots are complex if N is odd, and there is exactly one more real root, which is negative, when N is even. Furthermore, they are all irreducible (as a consequence of Theorem 2.2.5 in [16]). It can even be proved that \(\{\phi _N\}\) is a strictly increasing sequence with limit 2.

Recurrence sequences satisfying some N-bonacci recurrence relation are sometimes called generalized Fibonacci numbers or N-bonacci numbers (we have the Tetranacci numbers, A000288 in [18], the Pentanacci numbers, A000322, and so forth). Since the N-bonacci polynomial satisfies our hypotheses, our results also apply to Dirichlet series associated to N-bonacci sequences.

6 Values at Negative Integers

In [14], Navas proves that the values of the meromorphic continuation of the Fibonacci Dirichlet series at the negative integers that are not poles are rational numbers that can be written in terms of the Fibonacci and Lucas sequences. We ask now what can be said, in the general setting, about the values at negative integers of our meromorphic continuation.

Clearly, some of these negative integers may be poles: for example, the points \(-2m\), \(m\in {\mathbb {N}}\), for a recurrence sequence of degree 2 whose principal root is a totally positive unit (see Example 1), or the points \(-3m\), \(m\in {\mathbb {N}}\), for a recurrence sequence of degree 3 with two complex roots (see Example 2). At the points for which that is not the case, we can prove that the values of the function are once again rational numbers.

Let us keep the same assumptions as in Sect. 4, that is, \(\{a_n\}\) a LRS of integers which has an irreducible minimal polynomial \(P(x)\in {\mathbb {Z}}[x]\) with a positive dominant root, and assume (once again, without loss of generality for our purposes) that \(\{a_n\}\) is strictly increasing and positive. If \(\alpha _1,\ldots ,\alpha _r\) are the roots of P(x) (ordered with decreasing modulus), we have

$$\begin{aligned} a_n=\lambda _1\alpha _1^n+\cdots +\lambda _r\alpha _r^n \quad \forall n\in {\mathbb {N}}\end{aligned}$$

for unique \(\lambda _1,\ldots ,\lambda _r\) in the splitting field of P(x), and the function \(\varphi (s)\) defined as in (4.5) represents a meromorphic continuation of the Dirichlet series \(\sum a_n^{-s}\) to the whole s-plane.

In the following proof we will use multi-index notation, so we introduce it here. A multi-index of length r is an element \(\varvec{\beta }=(\beta _1,\ldots ,\beta _r)\in {\mathbb {N}}_0^r\). Given \(\varvec{\beta }\), we write:

  • \(|\varvec{\beta }|=\beta _1+\cdots +\beta _r\).

  • \(\displaystyle \left( {\begin{array}{c}|\varvec{\beta }|\\ \varvec{\beta }\end{array}}\right) =\frac{|\varvec{\beta }|!}{\beta _1!...\beta _r!}\).

  • Given \(\varvec{z}=(z_1,\ldots ,z_r)\in {\mathbb {C}}^r\), \({\varvec{z}}^{\varvec{\beta }}=z_1^{\beta _1}...z_r^{\beta _r}\).

Theorem 6.1

With our previous hypothesis, if \(m\in {\mathbb {N}}\) verifies that \(s=-m\) is not a pole of \(\varphi (s)\), then \(\varphi (-m)\in {\mathbb {Q}}\).

Proof

For \(m\in {\mathbb {N}}\), the binomial coefficient

$$\begin{aligned} \left( {\begin{array}{c}-(-m)\\ k_1\end{array}}\right) =\left( {\begin{array}{c}m\\ k_1\end{array}}\right) \end{aligned}$$

is nonzero only if \(k_1\le m\), so the sum in \(k_1\) that appears in the expression (4.5) becomes a finite sum, that is,

$$\begin{aligned} \varphi (-m)=\sum _{k_{\le r-1=0}}^{m,k_{\le r-2}}\Lambda _{k_{\le r-1}}(-m)\frac{\alpha _1^{m-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}{1-\alpha _1^{m-k_1}\alpha _2^{k_1-k_2}...\alpha _r^{k_{r-1}}}. \end{aligned}$$

Since the sum of the exponents of the \(\alpha _i\) (and those of the \(\lambda _i\) in the full expansion of \(\Lambda _{k_{\le r-1}}(-m)\)) equals m in every term, this expression can be written more cleanly using multi-indices,

$$\begin{aligned} \varphi (-m)=\sum _{|\varvec{\beta }|=m}\left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) {\varvec{\lambda }}^{\varvec{\beta }}\frac{{\varvec{\alpha }}^{\varvec{\beta }}}{1-{\varvec{\alpha }}^{\varvec{\beta }}}, \end{aligned}$$
(6.2)

where \(\varvec{\lambda }=(\lambda _1,\ldots ,\lambda _r)\) and \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _r)\).

We will show that (6.2) is an element of \({\mathbb {Q}}(\alpha _1,\ldots ,\alpha _r)^{{\mathcal {S}}_r}\), that is, a rational function of \(\alpha _1,\ldots ,\alpha _r\) invariant under the action of the symmetric group \({\mathcal {S}}_r\). This means in particular that it is an element of the decomposition field of P(x) invariant under its Galois group, so it is a rational number.

We can write

$$\begin{aligned} \varphi (-m)=\frac{\sum _{|\varvec{\beta }|=m}\left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) (\varvec{\lambda }\varvec{\alpha })^{\varvec{\beta }} \underset{\varvec{\beta }'\ne \varvec{\beta }}{\prod _{|\varvec{\beta }'|=m}} (1-{\varvec{\alpha }}^{\varvec{\beta }'})}{\prod _{|\varvec{\beta }|=m}(1-{\varvec{\alpha }}^{\varvec{\beta }})} \end{aligned}$$
(6.3)

The denominator of this expression is evidently invariant under permutations of the \(\alpha _i\), so it is an integer. Since \(-m\) is not a pole of \(\varphi (s)\), it must be nonzero. It remains to see that the numerator in (6.3) is a rational number.

The main difficulty here is dealing with the \(\lambda _i\). Note, however, that from Binet’s formula we can obtain the matrix equation

$$\begin{aligned} \left( \begin{array}{ccc} \alpha _1 &{} \ldots &{} \alpha _r \\ \vdots &{} \ddots &{} \vdots \\ \alpha _1^r &{} \ldots &{} \alpha _r^r\end{array}\right) \left( \begin{array}{c} \lambda _1 \\ \vdots \\ \lambda _r\end{array}\right) = \left( \begin{array}{c} a_1 \\ \vdots \\ a_r\end{array}\right) , \end{aligned}$$

and using Cramer’s rule we get that each \(\lambda _i\) is

$$\begin{aligned} \lambda _i= \left| \begin{array}{ccccccc} \alpha _1 &{} \ldots &{} \alpha _{i-1} &{} a_1 &{} \alpha _{i+1} &{} \ldots &{} \alpha _r \\ \vdots &{} &{} \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ \alpha _1^r &{} \ldots &{} \alpha _{i-1}^r &{} a_r &{} \alpha _{i+1}^r &{} \ldots &{} \alpha _r^r\end{array}\right| \left| \begin{array}{ccc} \alpha _1 &{} \ldots &{} \alpha _r \\ \vdots &{} \ddots &{} \vdots \\ \alpha _1^r &{} \ldots &{} \alpha _r^r\end{array}\right| ^{-1}. \end{aligned}$$
(6.4)

The second of the determinants in (6.4) is clearly \(\alpha _1...\alpha _r{{\text {V}}}(\alpha _1,\ldots ,\alpha _r)\), if \({{\text {V}}}(...)\) is the Vandermonde determinant, and a simple computation gives us that the first is

$$\begin{aligned} (-1)^{i-1}\sum _{j=1}^r (-1)^ja_jM_{ij}, \end{aligned}$$

where \(M_{ij}\) is the determinant

$$\begin{aligned} M_{ij}=\left| \begin{array}{cccccc} \alpha _1 &{} \ldots &{} \alpha _{i-1} &{} \alpha _{i+1} &{} \ldots &{} \alpha _r \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ \alpha _1^{j-1} &{} \ldots &{} \alpha _{i-1}^{j-1} &{} \alpha _{i+1}^{j-1} &{} \ldots &{} \alpha _r^{j-1} \\ \alpha _1^{j+1} &{} \ldots &{} \alpha _{i-1}^{j+1} &{} \alpha _{i+1}^{j+1} &{} \ldots &{} \alpha _r^{j+1} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ \alpha _1^r &{} \ldots &{} \alpha _{i-1}^r &{} \alpha _{i+1}^r &{} \ldots &{} \alpha _r^r\end{array}\right| , \end{aligned}$$

that is, the Vandermonde determinant with the i-th column and the j-th row removed.

With this, equation (6.4) becomes

$$\begin{aligned} \lambda _i=(-1)^{i-1}\frac{\displaystyle \sum \nolimits _{j=1}^r (-1)^j a_j M_{ij}}{\alpha _1...\alpha _r{{\text {V}}}(\alpha _1,\ldots ,\alpha _r)}, \end{aligned}$$

so

$$\begin{aligned} {\varvec{\lambda }}^{\varvec{\beta }}=\prod _{i=1}^r\left( (-1)^{i-1}\frac{\sum _{j=1}^r (-1)^j a_j M_{ij}}{\alpha _1...\alpha _r{{\text {V}}}(\alpha _1,\ldots ,\alpha _r)}\right) ^{\beta _i} \end{aligned}$$

and the numerator of (6.3) is

$$\begin{aligned} \sum _{|\varvec{\beta }|=m}\left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) \left( \prod _{i=1}^r\left( (-1)^{i-1}\frac{\sum _{j=1}^r (-1)^j a_j M_{ij}}{{{\text {V}}}(\alpha _1,\ldots ,\alpha _r)}\right) ^{\beta _i}\right) \underset{\varvec{\beta }'\ne \varvec{\beta }}{\prod _{|\varvec{\beta }'|=m}}(1-{\varvec{\alpha }}^{\varvec{\beta }'}). \end{aligned}$$
(6.5)

Let us check that this is invariant under permutations of the \(\alpha _i\). Clearly, it suffices to check that it is invariant under transpositions, so let \(\tau _{\gamma \delta }\), with \(\gamma <\delta \), be the trasposition that interchanges \(\alpha _{\gamma }\) and \(\alpha _{\delta }\).

First, it is trivial to see that

$$\begin{aligned} \tau _{\gamma \delta }({{\text {V}}}(\alpha _1,\ldots ,\alpha _r))=(-1){{\text {V}}}(\alpha _1,\ldots ,\alpha _r). \end{aligned}$$

Second, for the determinants \(M_{ij}\), we get that

$$\begin{aligned} \tau _{\gamma \delta }(M_{ij})=(-1)M_{ij}\quad \text { if }i\ne \gamma ,\delta , \end{aligned}$$

and that

$$\begin{aligned} \tau _{\gamma \delta }(M_{\gamma j})=(-1)^{\delta -\gamma -1}M_{\delta j}, \qquad \tau _{\gamma \delta }(M_{\delta j})=(-1)^{\delta -\gamma -1} M_{\gamma j}. \end{aligned}$$

As a consequence, in the product

$$\begin{aligned} {\varvec{\lambda }}^{\varvec{\beta }}{\varvec{\alpha }}^{\varvec{\beta }}=\prod _{i=1}^r\left( (-1)^{i-1}\frac{\sum _{j=1}^r (-1)^j a_j M_{ij}}{{{\text {V}}}(\alpha _1,\ldots ,\alpha _r)}\right) ^{\beta _i}, \end{aligned}$$

\(\tau _{\gamma \delta }\) does not change the factors with \(i\ne \gamma ,\delta \), and it interchanges the bases of the factors with \(i=\gamma \) and \(i=\delta \). Therefore, the effect of \(\tau _{\gamma \delta }\) on \((\varvec{\lambda }\varvec{\alpha })^{\varvec{\beta }}\) is to change

$$\begin{aligned} \varvec{\beta }=(\beta _1,\ldots ,\beta _{\gamma },\ldots \beta _{\delta },\ldots \beta _r) \end{aligned}$$

into

$$\begin{aligned} (\beta _1,\ldots ,\beta _{\delta },\ldots ,\beta _{\gamma },\ldots ,\beta _r). \end{aligned}$$

Checking now what happens to each term of the sum in (6.5), if we denote by \(\tau _{\gamma \delta }(\varvec{\beta })\) the result of interchanging the \(\gamma \)-th and \(\delta \)-th components, we get that

$$\begin{aligned} \tau _{\gamma \delta }\left( \left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) (\varvec{\lambda }\varvec{\alpha })^{\varvec{\beta }}\underset{\varvec{\beta }'\ne \varvec{\beta }}{\prod _{|\varvec{\beta }'|=m}}(1-{\varvec{\alpha }}^{\varvec{\beta }'})\right) =\left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) (\varvec{\lambda }\varvec{\alpha })^{\tau _{\gamma \delta }(\varvec{\beta })}\underset{\varvec{\beta }'\ne \varvec{\beta }}{\prod _{|\varvec{\beta }'|=m}}(1-{\varvec{\alpha }}^{\tau _{\gamma \delta }(\varvec{\beta }')}). \end{aligned}$$

Since

$$\begin{aligned} \left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) =\left( {\begin{array}{c}m\\ \tau _{\gamma \delta }(\varvec{\beta })\end{array}}\right) , \end{aligned}$$

we find that the effect of \(\tau _{\gamma \delta }\) in the full sum

$$\begin{aligned} \sum _{|\varvec{\beta }|=m}\left( {\begin{array}{c}m\\ \varvec{\beta }\end{array}}\right) (\varvec{\lambda }\varvec{\alpha })^{\varvec{\beta }}\underset{\varvec{\beta }'\ne \varvec{\beta }}{\prod _{|\varvec{\beta }'|=m}}(1-{\varvec{\alpha }}^{\varvec{\beta }'}) \end{aligned}$$

is to just commute its terms, which leaves the sum invariant. Therefore, we get that the numerator in (6.3) is also invariant under permutations of the \(\alpha _i\), which means it is a rational number. \(\square \)

7 Closing Remarks

Even if we are mainly interested in studying integer LRS whose minimal polynomial is irreducible, many of our results are still true if we remove some of these restrictions. For example, the proof of our main result about meromorphic continuation, Theorem 4.1, can be carried in exactly the same way if, instead of an integer LRS with irreducible minimal polynomial, we relax the hypotheses to allow for a real recurrence sequence with a dominant root \(>1\) with multiplicity 1 (there can be multiple roots as long as they are not the dominant one). Corollary 4.7 holds under the same hypotheses.

Results such as Mignotte’s theorem (Theorem 5.1) may be applied similarly as we have done in Sect. 5 to prove the distinctness of the poles (4.2) in some other cases of general degree.

As for Theorem 6.1 about the values of the zeta function at negative integers which are not poles, once again the same proof holds if we assume the sequence to be only rational, not integer, but still with a separable (not necessarily irreducible) minimal polynomial and with a dominant root \(>1\).

Finally, we wish to thank the anonymous referee for their comments and in particular for pointing out Mignotte’s result [12] on Pisot numbers, which we were not aware of.