A necessary and sufficient criterion for the existence of ratio limits of sequences generated by linear recurrences

TL;DR

This paper establishes a precise criterion to determine when the ratio limits of sequences generated by linear recurrences exist and what their values are, applicable to complex sequences.

Contribution

It provides a necessary and sufficient condition for the existence and value of ratio limits in sequences from arbitrary linear recurrences.

Findings

Derived a criterion for ratio limit existence

Applicable to complex sequences from linear recurrences

Clarified conditions for ratio limit values

Abstract

We introduce a necessary and sufficient criterion for determining the existence and the values of ratio limits of complex sequences generated by arbitrary linear recurrences.

Equations

$$P={\lambda }^n-b_1{\lambda }^{n-1}-\cdots -b_n,b_n\ne 0.$$

$$a_0{\lambda }_0^{n-1}+\sum _{i=1}^{n-1}{a}_{-i}\sum _{j=1}^{n-i}{b}_{i+j}{\lambda }_0^{n-j-1}\ne 0,$$

$$\text{then}\underset{k_0<k\to \infty }{\lim }\frac{F_{k+1}^{\mathbf{a}}}{F_k^{\mathbf{a}}}={\lambda }_0,F_k^{\mathbf{a}}\ne 0\text{for}k>k_0,$$

$$\text{where}{F}_k^{\mathbf{a}}=a_k\text{if}-n+1\le k\le 0\text{and}{F}_k^{\mathbf{a}}=b_1{F}_{k-1}^{\mathbf{a}}+\cdots +b_n{F}_{k-n}^{\mathbf{a}}\text{if}k>0.$$

$$\underset{k_0<k\to \infty }{\lim }\frac{F_{k+1}^{\mathbf{a}}}{F_k^{\mathbf{a}}}=\Psi ,\mathrm{where}{F}_k^{\mathbf{a}}\ne 0\mathrm{for}k>k_0,$$

$$f(z_{1},\dots,z_{n})=\biggr{(}z_{2},\dots,z_{n},b_{1}+\frac{b_{2}}{z_{n}}+\frac{b_{3}}{z_{n-1}z_{n}}+\dots+\frac{b_{n}}{z_{2}\dots z_{n}}\biggr{)}.$$

$$f\biggr{(}\frac{F_{k_{0}+2}}{F_{k_{0}+1}},\dots,\frac{F_{k_{0}+n+1}}{F_{k_{0}+n}}\biggr{)}\!=\biggr{(}\frac{F_{k_{0}+3}}{F_{k_{0}+2}},\dots,\frac{F_{k_{0}+n+1}}{F_{k_{0}+n}},\frac{b_{1}F_{k_{0}+n+1}+b_{2}F_{k_{0}+n}+\dots+b_{n}F_{k_{0}+2}}{F_{k_{0}+n+1}}\biggr{)}.$$

$$f\biggr{(}\frac{F_{k_{0}+2}}{F_{k_{0}+1}},\dots,\frac{F_{k_{0}+n+1}}{F_{k_{0}+n}}\biggr{)}=\biggr{(}\frac{F_{k_{0}+3}}{F_{k_{0}+2}},\dots,\frac{F_{k_{0}+n+2}}{F_{k_{0}+n+1}}\biggr{)}.$$

$$f(\Psi ,\dots ,\Psi )=(\Psi ,\dots ,\Psi ).$$

$$\underset{k_0\to \infty }{\lim }\frac{F_{k_0+i+1}}{F_{k_0+n+1}}={\Psi }^{-n+i},i=1,\dots ,n,$$

$$f(\Psi,\dots,\Psi)=\big{(}\Psi,\dots,\Psi,b_{1}+b_{2}\Psi^{-1}+\dots+b_{n}\Psi^{1-n}).$$

$$(k^j{\lambda }_i^k{)}_{k=-n+1}^{\infty },i=1,\dots ,\nu ,j=0,\dots ,{\mu }_i-1,$$

$$F_k^{\mathbf{a}}=\sum _{i=1}^{\nu }\sum _{j=0}^{{\mu }_i-1}{c}_{ij}^{\mathbf{a}}{k}^j{\lambda }_i^k.$$

$${\mathbf{c}}^{\mathbf{a}}=C^{-1}\mathbf{a},$$

$$C=\left[\begin{array}[]{ccccccc}1&\cdots&0&\cdots&1&\cdots&0\\ \lambda_{1}^{-1}&\cdots&(-1)^{\mu_{1}-1}\lambda_{1}^{-1}&\cdots&\lambda_{\nu}^{-1}&\cdots&(-1)^{\mu_{\nu}-1}\lambda_{\nu}^{-1}\\ \lambda_{1}^{-2}&\cdots&(-2)^{\mu_{1}-1}\lambda_{1}^{-2}&\cdots&\lambda_{\nu}^{-2}&\cdots&(-2)^{\mu_{\nu}-1}\lambda_{\nu}^{-2}\\ \vdots&\ddots&\vdots&\ddots&\vdots&\ddots&\vdots\\ \lambda_{1}^{-n+1}&\cdots&(-n+1)^{\mu_{1}-1}\lambda_{1}^{-n+1}&\cdots&\lambda_{\nu}^{-n+1}&\cdots&(-n+1)^{\mu_{\nu}-1}\lambda_{\nu}^{-n+1}\\ \end{array}\right].$$

$$\underset{k_0<k\to \infty }{\lim }\frac{F_{k+1}^{\mathbf{a}}}{F_k^{\mathbf{a}}},\mathrm{where}{F}_k^{\mathbf{a}}\ne 0\text{f}ork>k_0,$$

$$\underset{k_0<k\to \infty }{\lim }\frac{F_{k+1}^{\mathbf{a}}}{F_k^{\mathbf{a}}}={\lambda }_{i_0},$$

$$\underset{k\to \infty }{\lim }\frac{F_k^{\mathbf{a}}}{k^{j_0}{\lambda }_{i_0}^k}=c_{i_0{j}_0}.$$

$$\lim_{k_{0}<k\to\infty}\frac{F_{k+1}^{\bf a}}{F_{k}^{\bf a}}=\lambda_{i_{0}}\cdot\lim_{k_{0}<k\to\infty}\biggr{(}\frac{F_{k+1}^{\bf a}}{(k+1)^{j_{0}}\lambda_{i_{0}}^{k+1}}\cdot\frac{k^{j_{0}}\lambda_{i_{0}}^{k}}{F_{k}^{\bf a}}\biggr{)}=\lambda_{i_{0}}.$$

$$D_k^{\mathbf{a}}=k^{j_0}\sum _{l=1}^{\eta }{c}_{l{j}_0}^{\mathbf{a}}{\lambda }_l^k.$$

$$\tilde{\lambda }=\underset{k_0<k\to \infty }{\lim }\frac{F_{k+1}^{\mathbf{a}}}{F_k^{\mathbf{a}}}=\underset{k_0<k\to \infty }{\lim }\frac{D_{k+1}^{\mathbf{a}}+E_{k+1}^{\mathbf{a}}}{D_k^{\mathbf{a}}+E_k^{\mathbf{a}}}=\underset{k_0<k\to \infty }{\lim }\frac{D_{k+1}^{\mathbf{a}}}{D_k^{\mathbf{a}}+E_k^{\mathbf{a}}}.$$

$${\tilde{\lambda }}^{-1}=\underset{k_0<k\to \infty }{\lim }\frac{D_k^{\mathbf{a}}+E_k^{\mathbf{a}}}{D_{k+1}^{\mathbf{a}}}=\underset{k_0<k\to \infty }{\lim }\frac{D_k^{\mathbf{a}}}{D_{k+1}^{\mathbf{a}}}.$$

$$\lim_{k_{0}<k\to\infty}\biggr{(}\frac{\sum_{l=1}^{\eta}c^{\bf a}_{lj_{0}}(\lambda_{l}/R)^{k+1}}{\sum_{l=1}^{\eta}c^{\bf a}_{lj_{0}}(\lambda_{l}/R)^{k}}\biggr{)}=\tilde{\lambda}/R.$$

$$\biggr{(}\frac{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}}\biggr{)}^{\infty}_{k=k_{0}+1}$$

$$\biggr{|}\frac{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+2}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}}-\frac{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}}\biggr{|}<\epsilon.$$

$$\biggr{|}\frac{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+2}\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}-\big{(}\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}\big{)}^{2}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}}\biggr{|}=$$

$$=\biggr{|}\frac{\sum_{l=1}^{\eta}\sum_{m=l+1}^{\eta}c^{\bf a}_{l}c^{\bf a}_{m}\gamma^{k}_{l}\gamma^{k}_{m}(\gamma_{l}-\gamma_{m})^{2}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}}\biggr{|}<\epsilon.$$

$$\biggr{(}\frac{\sum_{l=1}^{\eta}\sum_{m=l+1}^{\eta}c^{\bf a}_{l}c^{\bf a}_{m}\gamma^{k}_{l}\gamma^{k}_{m}(\gamma_{l}-\gamma_{m})^{2}}{\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k+1}\sum_{l=1}^{\eta}c^{\bf a}_{l}\gamma_{l}^{k}}\biggr{)}^{\infty}_{k=k_{0}+1}$$

Full text

1. Introduction

Sloane’s Online Encyclopedia of Integer Sequences [22] and Khovanova’s website [16] catalog thousands of integer sequences generated by linear recurrences that are associated with problems in various branches of mathematics and other sciences, such as number theory, abstract algebra, linear algebra, combinatorics, complex numbers, group theory, probability, statistics, affine geometry, electrical networks, infectious diseases, etc., cf.  [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 17, 18, 20, 21, 23, 24, 25].

The asymptotic behavior of sequences generated by linear recurrences is characterized by the ratio limit of the sequence’s consecutive terms. The knowledge on whether a ratio limit exists is necessary if a problem requires considering the sequence’s terms with higher and higher indices. The existence of a ratio limit and its value depend on the choice of the sequence’s initial conditions.

In 1997 Dubeau et al. [9] studied linear recurrences $F\in L(\mathbb{C}^{n},\mathbb{C}^{n})$ with asymptotically simple characteristic polynomials

[TABLE]

A polynomial is asymptotically simple iff among its zeros of maximal modulus there is a dominant zero $\lambda_{0}$ of maximal multiplicity.

Dubeau et al. derived *a sufficient * criterion for the existence of ratio limits of sequences \big{(}F^{\bf a}_{k}\big{)}_{k=-n+1}^{\infty} generated by $F$ from complex initial conditions ${\bf a}=(a_{-n+1},\dots,a_{0})$. Specifically, the authors showed that if

[TABLE]

[TABLE]

[TABLE]

Condition (1.2) is satisfied, in particular, by all sequences generated by linear recurrences with asymptotically simple characteristic polynomials from initial conditions $(0,\dots,0,a_{0})$.

An example of a sequence that does not satisfy condition (1.2), but has the ratio limit, is the constant sequence $(1_{k})_{k=-n+1}^{\infty}$ generated by the linear recurrence with the signature $(2,-1)$ from the initial conditions $(1,1)$. The corresponding asymptotically simple characteristic polynomial $P=(\lambda-1)^{2}$​.

We generalize results obtained by Dubeau et al. by introducing *a necessary and sufficient * criterion for the existence of ratio limits of complex sequences generated by linear recurrences with *arbitrary * characteristic polynomials $P$. We also prove that if the ratio limit exists, it must be equal to one of the zeros of $P$.

2. Main results

Given a linear recurrence $F\in L(\mathbb{C}^{n},\mathbb{C}^{n})$ of an order $n$ with the signature $(b_{1},\dots,b_{n})$, where $b_{n}\!\neq\!0$. A sequence $(F^{\bf a}_{k})_{k=-n+1}^{\infty}=F^{\bf a}$ generated by formulas (1.4) is called a solution of $F$.

Theorem 2.1**.**

If a solution $F^{\bf a}$ of $F$ generated from initial conditions ${\bf a}\in\mathbb{C}^{n}$ has a ratio limit

[TABLE]

then $\Psi$ is equal to one of the zeros of the characteristic polynomial $P$ of $F$.

Proof.

If $n=1$, $b_{1}$ is the zero of the characteristic monomial, and we have $(F^{\bf a}_{k})_{k=0}^{\infty}=a_{1}\cdot b_{1}^{k}$.

If $n>1$, we introduce a continuous mapping $f:(\mathbb{C}^{*})^{n}\to(\mathbb{C}^{*})^{n}$ defined as

[TABLE]

[TABLE]

It follows from formula (1.4) with $k=k_{0}+n+2$, and equation (2.3) that

[TABLE]

Our assumption (2.1) and equation (2.4) imply that iterations of $f$ create a sequence convergent to the vector $(\Psi,\dots,\Psi)$. Since $f$ is continuous, it means that the vector $(\Psi,\dots,\Psi)$ is a fixed point of $f$, [19, p.227], i.e.,

[TABLE]

On the other hand, from the continuity of $f\!$, equation (2.3), and the fact that due to (2.1)

[TABLE]

we obtain that

[TABLE]

Equations (2.5) and (2.7) imply that $\,\Psi^{n}-b_{1}\Psi^{n-1}-\cdots-b_{n}=0$.

∎

Let the characteristic polynomial $P$ of a linear recurrence $F$ have $\nu$ distinct zeros. For simplicity of the notation, we label them as $\lambda_{i}$, $i=1,\dots,\nu$. Let $\mu_{i}$ denote the multiplicity of the zero $\lambda_{i}$, i.e., $\sum_{i=1}^{\nu}\mu_{i}=n$.

Any solution $F^{\bf a}$ of $F$ is a linear combination of the following $n$ basic solutions of $F$ [14, 15]:

[TABLE]

So, for $k\geq-n+1$, we have

[TABLE]

The coefficients $(c^{\bf a}_{1,0},\dots,c^{\bf a}_{\nu,\mu_{\nu}-1})={\bf c^{\bf a}}$ are solutions of the system of linear equations

[TABLE]

where columns of matrix $C$ consists of linearly independent vectors built from the initial conditions of basic solutions (2.8), i.e.,

[TABLE]

We define the characteristic polynomial $P^{\bf a}$ of a solution $F^{\bf a}$ as follows:

• •

$P^{\bf a}$ has as its zeros all those zeros $\lambda_{i}$ of $P$ for which there exists $j$ such that $c_{ij}^{\bf a}\neq 0$;

• •

The multiplicity of a zero $\lambda_{i}$ in $P^{\bf a}$ is equal to the largest index $j$ for which $c_{ij}^{\bf a}\neq 0$.

In what follows, we say that a solution $F^{\bf a}$ of a linear recurrence $F$ *is associated * with the characteristic polynomial $P^{\bf a}$​.

Theorem 2.2**.**

Given a solution $F^{\bf a}$ of a linear recurrence $F$. The ratio limit

[TABLE]

exists iff the characteristic polynomial $P^{\bf a}$ of the solution $F^{\bf a}$ is asymptotically simple.

If the latter is true, then

[TABLE]

where $\lambda_{i_{0}}$ is the dominant zero of $P^{\bf a}\!$.111Condition (1.2) ensures that the dominant zero $\lambda_{i_{0}}$ of $P^{\bf a}\neq P$ coincides with the dominant zero $\lambda_{0}$ of $P$.

Proof.

$\Longleftarrow$ Let $\lambda_{i_{0}}$ be the dominant zero with the multiplicity $j_{0}$ of the asymptotically simple polynomial $P^{\bf a}\!$. Then, we have

[TABLE]

Formula (2.13) implies that

[TABLE]

$\Longrightarrow$ Let us assume that the ratio limit exists and that the characteristic polynomial $P^{\bf a}$ of a solution $F^{\bf a}$ is not asymptotically simple. Then $P^{\bf a}$ has $\eta>1$ *distinct * zeros, say $\lambda_{1},\dots,\lambda_{\eta}$, such that

1. (i)

the modulus $R=|\lambda_{1}|=\dots=|\lambda_{\eta}|\,$ is greater than or equal to the moduli of other zeros of the polynomial $P^{\bf a}$; and 2. (ii)

there exist nonzero coefficients $c^{\bf a}_{1j_{0}},\dots,c^{\bf a}_{\eta j_{0}}$ with the index $j_{0}$ greater than all indices $j$ corresponding to zeros of $P^{\bf a}$ with the same modulus as $R$.

We decompose each sequence element $F^{\bf a}_{k}$ given by formula (2.9) into a part $D^{\bf a}_{k}$ containing linear combinations of the basic solutions (2.8) with the dominant moduli equal to $|k^{j_{0}}R^{k}|$, and a part $E^{\bf a}_{k}$ containing linear combinations of the basic solutions with moduli smaller than $|k^{j_{0}}R^{k}|$. Thus, for any $k\geq-n+1$, we have $F^{\bf a}_{k}=D^{\bf a}_{k}+E^{\bf a}_{k}$, where

[TABLE]

According to Theorem 2.1, if limit (2.11) exists, it is equal to a zero of the characteristic polynomial $P\!$, say $\tilde{\lambda}$. So, we obtain that

[TABLE]

Formula (2.16) implies that

[TABLE]

It follows from (2.15) and (2.17) that

[TABLE]

To simplify the notation, let us set $c^{\bf a}_{l}=c^{\bf a}_{lj_{0}}$, and let us introduce normalized zeros $\gamma_{l}=\lambda_{l}/R$, i.e., $|\gamma_{l}|=1$, $l=1,\dots,\eta$. Since limit (2.18) exists, the sequence

[TABLE]

is a Cauchy sequence. Thus, for any $\epsilon>0$, there exist $k_{\epsilon}$ such that for $k>k_{\epsilon}$

[TABLE]

We transform inequality (2.20) into

[TABLE]

[TABLE]

It follows from inequality (2.22) that the sequence

[TABLE]

must converge to 0.

The denominators in sequence (2.23) are bounded from above due to the fact that the moduli $|\gamma_{l}|=1$, $l=1,\dots,\eta$. Thus, the numerators of this sequence must form a sequence converging to 0. However, the sequences $(\gamma^{k}_{l}\gamma^{k}_{m})_{k=k_{0}+1}^{\infty}$ oscillate for each pair of indices $(l,m)$, and therefore their linear combination does not converge to 0. Consequently, sequence (2.23) converges to 0 only when it is the constant sequence $(0)^{\infty}_{k=k_{0}+1}$, i.e., all normalized zeros $\gamma_{l}$ are equal one to another.

So, if the ratio limit (2.11) exists, there can be only one zero that satisfies conditions (i) and (ii) listed above. The latter contradicts our assumption that the characteristic polynomial $P^{\bf a}$ of the solution $F^{\bf a}$ is not asymptotically simple.

∎