# On solutions for some class of integrable difference equations

Andrei K. Svinin  Andrei K. Svinin, Matrosov Institute for System Dynamics and
Control Theory of Siberian Branch of Russian Academy of Sciences, P.O. Box
292, 664033 Irkutsk, Russia<EMAIL_ADDRESS>
###### Abstract.

In this paper we show that an arbitrary solution of one ordinary difference
equation is also a solution for a hierarchy of integrable difference
equations. We also provide an example of such a solution that is related to
sequence generated by a second-order linear recursion with 2-periodic
coefficients.

##  1\. Introduction

In this paper we consider an infinite one-parameter class of ordinary
difference equations

$y_{n+1}\left(\sum_{j=0}^{s-1}y_{n+j}-H^{(s)}\right)=y_{n+s}\left(\sum_{j=0}^{s-1}y_{n+j+2}-H^{(s)}\right)\;\;s\geq
2.$ (1.1)

Notice that each of equations (1.1) contains one arbitrary parameter $H^{(s)}$
and can be rewritten in the form

$y_{n+s+1}=R(y_{n},\ldots,y_{n+s};H^{(s)}),$

where $R$ is corresponding rational function of its arguments. Equation (1.1),
for some fixed value of $H^{(s)}$, yields a map
$\mathbb{R}^{s+1}\rightarrow\mathbb{R}^{s+1}$ for real-valued initial data
$\\{y_{n}:n=1,\ldots,s\\}$. By analogy with ordinary differential equations, a
function $J=J_{n}=J(y_{n},\ldots,y_{n+s})$ is called a first integral for
difference equation (1.1) if by virtue of this equation one has
$J_{n+1}=J_{n}$. There are ordinary difference equations which have some
special properties which yield a regular behavior of their solutions. One such
property is Liouville integrability [2], [17].

Equations of the form (1.1) appeared in [14]. They are interesting in that
they share the property of having a Lax pair that indicates their possible
Liouville integrability. In [15], we have shown a method for constructing a
set of the first integrals for these equations and also for a wider class of
integrable difference equation regardless of their Lax representation.
Unfortunately, the property of Liouville integrability for these equations has
not been proved in our works. We say that these equations are integrable only
in the sense that they admit a nontrivial Lax representation that yields a
number of the first integrals.

Also note that, as was shown in [14] that, assuming that $y_{n}$’s depend on
evolutionary parameters $(t_{1},t_{2},\ldots)$, equations (1.1) play the role
of compatible constraints for the Volterra lattice

$\frac{\partial y_{n}}{\partial t_{1}}=y_{n}\left(y_{n+1}-y_{n-1}\right)$

and its hierarchy of generalized symmetries that can be written in explicit
form as [14]

$\frac{\partial y_{n}}{\partial
t_{s}}=y_{n}\left(S^{s}_{s}(n-s+2)-S^{s}_{s}(n-s)\right)\;\;s\geq 2.$ (1.2)

Here discrete polynomials $S_{s}^{k}(n)$ are explicitly defined by

$S_{s}^{k}(n)=\sum_{0\leq q_{1}\leq\cdots\leq q_{k}\leq
s-1}y_{n+q_{1}+k-1}\cdots y_{n+q_{k}}.$ (1.3)

It should be noted that the first integrals constructed in [15] are described
in terms of these discrete polynomials.

In this article we do not discuss Liouville integrability for equations (1.1).
Instead, we show a compatibility of these equations, which also, to a certain
extent, reflects the integrability property. In general, it should be said
that the integrability property for equations of various nature is often
formulated as a compatibility condition of some number of equations. An
example is the pairwise compatibility of evolutionary equations (1.2).

Specifically, we prove the fact that an arbitrary solution of equation (1.1)
corresponding to the simplest case $s=2$, that is,

$y_{n+1}\left(y_{n}+y_{n+1}-H^{(2)}\right)=y_{n+2}\left(y_{n+2}+y_{n+3}-H^{(2)}\right)$
(1.4)

is also a solution of equation (1.1) for all $s\geq 3$. In these
circumstances, it turns out that parameters $H^{(s)}$ are related to each
other by rather nontrivial recurrence relations which also include two first
integrals $G^{(s)}$ and $G_{1}^{(s)}$ that will be defined in the following
section. Ultimately each $H^{(s)}$, for $s\geq 3$, is unambiguously expressed
as a rational function of variables $(y_{0},y_{1},y_{2},H^{(2)})$.

Also in the paper we construct some class of solutions for (1.1) that are
yielded by a second-order linear recursion with 2-periodic coefficients

$T_{n}=P_{n}T_{n-1}+QT_{n-2},\;\;P_{n+2}=P_{n}$ (1.5)

with arbitrary initial conditions. Rather important circumstance here is that
recursion (1.5) defines a great number of interesting integer sequences with
applications in various areas of mathematics.

The paper is organized as follows. In the following section, using the two
integrals mentioned above and some substitutions, we show some difference
equations closely related to (1.1) which are of interest in themselves. In
particular, [10] we show that equations (1.1) are related to bilinear
difference equations with 2-periodic coefficients. It should be noted that
presentation of material in Section 2 partially follows the lines of the paper
[10]. At the end of Section 2, we formulate a theorem concerning the above-
mentioned compatibility of equations (1.1). In Section 3, we show how the
sequences defined by recursion (1.5) yield solutions for equation (1.4) and,
therefore, for all equations of the form (1.1).

##  2\. Difference equations

To begin with, note that equation (1.1) has the first integral

$G^{(s)}=\left(\sum_{j=0}^{s}y_{n+j}-H^{(s)}\right)\prod_{j=1}^{s-1}y_{n+j}.$
(2.1)

We can consider this relation as a difference equation of order $s$ with two
parameters $G^{(s)}$ and $H^{(s)}$, that is equivalent to (1.1). Indeed, it is
easy to see that the relation $G^{(s)}(n+1)=G^{(s)}(n)$ is nothing but (1.1).

######  Remark 2.1.

The equation (2.1) known as the generalized DTKQ equation was the object of
study in the work [10]. It was shown there that for even $s$ the equation
(2.1) are related to the $(s,-1)$ periodic reduction of Hirota’s discrete KdV
equation, that turned out to be Liouville integrable [7]. In turn, for odd
$s$, equations (2.1) are connected to reductions of a discrete Toda lattice
[9]. It should be also noticed, that this equation with $H^{(s)}=0$ has
appeared in [3].

More general than (2.1) is a two-parameter class of integrable equations

$G^{(k,s)}=\sum_{j=0}^{k}(-1)^{j}H_{j}^{(k,s)}S^{k-j}_{s-j+1}(n+j)\prod_{j=k}^{s-1}y_{n+j},\;\;k\geq
1,\;\;s\geq k+1,$

where $H_{0}^{(k,s)}=1$, appeared in [14] and [15]. Here $(k,s)$-th difference
equation involves $k+1$ parameters
$\left(G^{(k,s)},H_{1}^{(k,s)},\ldots,H_{k}^{(k,s)}\right)$.

Let us consider the first integral [10], [15]

$G_{1}^{(s)}=\prod_{j=0}^{s}y_{n+j}+\sum_{j=1}^{s-1}y_{n+j}\left(\sum_{j=0}^{s}y_{n+j}-H^{(s)}\right)\prod_{j=1}^{s-1}y_{n+j},$

which further will be useful for us. In what follows, we use the fact, that
$G_{1}^{(s)}$ can be rewritten as

$G_{1}^{(s)}=\sum_{j=0}^{s-1}y_{n+j}\left(G^{(s)}-\prod_{j=0}^{s-1}y_{n+j}\right)+H^{(s)}\prod_{j=0}^{s-1}y_{n+j}.$
(2.2)

With the substitution $y_{n}=w_{n}w_{n+1}$ equation (2.1) becomes

$\sum_{j=0}^{s}w_{n+j}w_{n+j+1}=\frac{G^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}}+H^{(s)}.$
(2.3)

By direct calculation, it can be checked that this equation has $2$-integral
[10]

$\alpha_{n}^{(s)}=\prod_{j=0}^{s}w_{n+j}-\frac{G^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}.$
(2.4)

Indeed, by (2.3), we have

$\alpha_{n+1}^{(s)}=\prod_{j=1}^{s-1}w_{n+j}\left(H^{(s)}-\sum_{j=0}^{s-1}w_{n+j}w_{n+j+1}\right)\;\;\mbox{and}\;\;\alpha_{n+2}^{(s)}=\alpha_{n}^{(s)}.$
(2.5)

######  Lemma 2.2.

The relation

$\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}=G_{1}^{(s)}-H^{(s)}G^{(s)}$ (2.6)

is valid.

Proof. By (2.4) and (2.5), we have

$\displaystyle\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}$ $\displaystyle=$
$\displaystyle\left(\prod_{j=0}^{s}w_{n+j}-\frac{G^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}\right)\prod_{j=1}^{s-1}w_{n+j}\left(H^{(s)}-\sum_{j=0}^{s-1}w_{n+j}w_{n+j+1}\right)$
(2.7) $\displaystyle=$
$\displaystyle\left(\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}-G^{(s)}\right)\left(H^{(s)}-\sum_{j=0}^{s-1}w_{n+j}w_{n+j+1}\right).$

On the other hand, with the substitution $y_{n}=w_{n}w_{n+1}$, (2.2) becomes

$G_{1}^{(s)}=\sum_{j=0}^{s-1}w_{n+j}w_{n+j+1}\left(G^{(s)}-\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}\right)+H^{(s)}\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}.$

Comparing the latter with (2.7), in a result, we get relation (2.6). $\Box$

Let us now rewrite (2.4) as

$\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}=\alpha_{n}^{(s)}\prod_{j=1}^{s-1}w_{n+j}+G^{(s)},\;\;\alpha_{n+2}^{(s)}=\alpha_{n}^{(s)}.$
(2.8)

This is an ordinary difference equation of order $s$. This equation is known
as the modified generalized DTKQ equation [10]. In particular case $s=2$, it
becomes

$w_{n}w_{n+1}^{2}w_{n+2}=\alpha_{n}^{(2)}w_{n+1}+G^{(2)},\;\;\alpha^{(2)}_{n+2}=\alpha^{(2)}_{n}.$
(2.9)

Equation (2.8) is of interest, because, as was pointed out in [10], it
represents an example of $U$-system [8].

Let us notice that here $H^{(s)}$ plays the role of the first integral for
difference equation (2.8). From (2.5), we derive the following expression for
this integral:

$H^{(s)}=\sum_{j=0}^{s-1}w_{n+j}w_{n+j+1}+\frac{\alpha_{n+1}^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}.$
(2.10)

By direct calculation, it can be checked that, by (2.8), the first integral
(2.10) can be rewritten as

$\displaystyle H^{(s)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{s-2}w_{n+j}w_{n+j+1}+\frac{\alpha_{n}^{(s)}}{\prod_{j=0}^{s-2}w_{n+j}}+\frac{\alpha_{n+1}^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}$
(2.11) $\displaystyle+\frac{G^{(s)}}{\prod_{j=0}^{s-2}w_{n+j}w_{n+j+1}}.$

In particular, in the case $s=2$, (2.11) becomes

$H^{(2)}=w_{n}w_{n+1}+\frac{\alpha_{n}^{(2)}}{w_{n}}+\frac{\alpha_{n+1}^{(2)}}{w_{n+1}}+\frac{G^{(2)}}{w_{n}w_{n+1}}.$
(2.12)

With the substitution $w_{n}=f_{n}f_{n+2}/f_{n+1}^{2}$ equation (2.8) can be
rewritten in bilinear form [10]

$f_{n}f_{n+s+2}=\alpha_{n}^{(s)}f_{n+1}f_{n+s+1}+G^{(s)}f_{n+2}f_{n+s},\;\;\alpha^{(s)}_{n+2}=\alpha^{(s)}_{n}.$
(2.13)

More exactly, any solution of (2.13) yields a solution of (2.8) but not vice
versa. In particular, if $s=2$, then (2.13) becomes

$f_{n}f_{n+4}=\alpha_{n}^{(2)}f_{n+1}f_{n+3}+G^{(2)}f_{n+2}^{2},\;\;\alpha^{(2)}_{n+2}=\alpha^{(2)}_{n}.$
(2.14)

######  Remark 2.3.

In fairness, we note that the substitution
$y_{n}=f_{n}f_{n+3}/(f_{n+1}f_{n+2})$ was used in [16] to put evolution
equations of the Volterra lattice hierarchy (1.2) into bilinear form.

######  Remark 2.4.

According to Theorem 1.1 from [10], the identities

$f_{n}f_{n+2s+1}=-G^{(s)}f_{n+1}f_{n+2s}+K^{(s)}f_{n+s}f_{n+s+1}\;\forall\;\;\mbox{even}\;s\geq
2$

and

$f_{n}f_{n+2s+2}=\left(G^{(s)}\right)^{2}f_{n+2}f_{n+2s}+K^{(s)}f^{2}_{n+s+1}\;\forall\;\;\mbox{odd}\;s\geq
3$

are valid, where $K^{(s)}$ is the first integral of $s$-th equation (2.1)
defined by [10]

$K^{(s)}=\left(\prod_{j=0}^{s-1}y_{n+2j}+G^{(s)}\right)\prod_{q=0}^{(s-2)/2}\prod_{j=1}^{s-1}y_{n+2q+j},\;\mbox{if}\;s\;\mbox{even}$

and

$K^{(s)}=\left(\prod_{j=0}^{2s-1}y_{n+j}-\left(G^{(s)}\right)^{2}\right)\prod_{q=0}^{(s-3)/2}\prod_{j=2}^{s}y_{n+2q+j},\;\mbox{if}\;s\;\mbox{odd}.$

######  Remark 2.5.

Equation (2.14) represents a slight generalization of the Somos-$4$ equation

$f_{n}f_{n+4}=f_{n+1}f_{n+3}+f_{n+2}^{2},$ (2.15)

that generate, for example, the sequence of positive integers that begins with
$(1,1,1,1,2,3,7,23,59,\ldots)$. Despite the fact that, generally speaking,
this recurrence relation must generate a sequence of fractional numbers, they
all turns out to be integers. A proof of integrity of this sequence known as
the Somos-4 sequence can be found in [6]. It has now become clear that the
reason for integrity of this sequence lies in the fact that equation (2.15)
has the Laurent property. In this connection, it is appropriate to mention
here the relationship of equation (2.15) to cluster algebras [4], [5]. One of
the properties of equation (2.13) is that it has also the Laurent property
(cf. [10]), that means that
$f_{n}\in\mathbb{Z}[\alpha_{0}^{(s)},\alpha_{1}^{(s)},f_{0}^{\pm
1},\ldots,f_{s+1}^{\pm 1}]$ for any $n\geq s+2$. The Laurent property for
equation (2.13), in fact, follows, as a special case, from this property for
the Hirota-Miwa equation [12].

In the sequel we need the following lemma.

######  Lemma 2.6.

Any solution $\\{w_{n}\\}$ to equation (2.9) is also a solution of the
equation

$w_{n}w_{n+1}^{2}w_{n+2}^{2}w_{n+3}=\alpha_{n}^{(3)}w_{n+1}w_{n+2}+G^{(3)}$
(2.16)

with

$\alpha_{0}^{(3)}=\alpha_{1}^{(3)}=-G^{(2)}\;\;\mbox{and}\;\;G^{(3)}=G_{1}^{(2)}=\alpha_{0}^{(2)}\alpha_{1}^{(2)}+G^{(2)}H^{(2)}.$
(2.17)

In addition, in these circumstances, we have $H^{(3)}=H^{(2)}$.

Proof. Provided that $\\{w_{n}\\}$ is any solution of equation (2.9), for some
values of $\alpha_{0}^{(2)},\alpha_{1}^{(2)}$ and $G^{(2)}$, we get the
following:

$\displaystyle w_{n}w_{n+1}^{2}w_{n+2}\cdot w_{n+1}w_{n+2}^{2}w_{n+3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left(\alpha_{n}^{(2)}w_{n+1}+G^{(2)}\right)\left(\alpha_{n+1}^{(2)}w_{n+2}+G^{(2)}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\alpha_{n}^{(2)}\alpha_{n+1}^{(2)}w_{n+1}w_{n+2}+G^{(2)}\left(\alpha_{n}^{(2)}w_{n+1}+\alpha_{n+1}^{(2)}w_{n+2}\right)+\left(G^{(2)}\right)^{2}.$

Taking into account (2.12), we get

$w_{n}w_{n+1}^{2}w_{n+2}^{2}w_{n+3}=-G^{(2)}w_{n+1}w_{n+2}+\left(\alpha_{n}^{(2)}\alpha_{n+1}^{(2)}+G^{(2)}H^{(2)}\right).$

Whence it follows that if $\\{w_{n}\\}$ satisfies equation (2.9), then it also
satisfies (2.16) with corresponding coefficients given by (2.17).

Moreover, by (2.9) and (2.10), we have

$\displaystyle H^{(3)}-H^{(2)}$ $\displaystyle=$ $\displaystyle
w_{n+2}w_{n+3}+\frac{\alpha^{(3)}_{n+1}}{w_{n+1}w_{n+2}}-\frac{\alpha^{(2)}_{n+1}}{w_{n+1}}$
$\displaystyle=$
$\displaystyle\left(\frac{\alpha^{(2)}_{n+1}}{w_{n+1}}+\frac{G^{(2)}}{w_{n+1}w_{n+2}}\right)+\frac{\alpha^{(3)}_{n+1}}{w_{n+1}w_{n+2}}-\frac{\alpha^{(2)}_{n+1}}{w_{n+1}}.$

The latter is zero by (2.17). Therefore this lemma is proved. $\Box$

Let us notice that equation (2.16) is not a simple consequence of equation
(2.9), as it may seem at the first glance, since $G^{(3)}$ depends not only on
coefficients of equation (2.9) but also on $H^{(2)}$, that is, these
coefficients depend on the solution itself. Also, it should be noted that
$\alpha_{n}^{(3)}$ does not depend on $n$ provided that $\\{w_{n}\\}$ is a
solution of equation (2.9).

######  Theorem 2.7.

Any solution $\\{w_{n}\\}$ to equation (2.9) is also a solution to equation
(2.8) for any $s\geq 4$ with suitable coefficients
$\alpha_{0}^{(s)},\alpha_{1}^{(s)}$ and $G^{(s)}$. These coefficients are
defined by the following recurrent relations:

$\alpha_{n}^{(s+2)}=\frac{G^{(s+1)}}{G^{(s)}}\alpha_{n}^{(s)}\;\;\mbox{and}\;\;G^{(s+2)}=G_{1}^{(s+1)}-\frac{G^{(s+1)}}{G^{(s)}}G_{1}^{(s)}\;\;s\geq
2,$ (2.18)

where $G_{1}^{(s)}=G^{(s)}H^{(s)}+\alpha_{0}^{(s)}\alpha_{1}^{(s)}$. In
addition, the first integrals $H^{(s)}$ are defined by the recurrent relation

$H^{(s+2)}=H^{(s+1)}+\frac{G^{(s+1)}}{G^{(s)}}\;\;s\geq 2.$ (2.19)

The proof of this theorem is quite technical and cumbersome and therefore we
carried it to the Appendix. Evidently recurrent relations (2.18) and (2.19)
supplemented by (2.17) unambiguously define coefficients
$\alpha_{0}^{(s)},\alpha_{1}^{(s)},G^{(s)}$, for all $s\geq 4$ as rational
functions of $\alpha_{0}^{(2)},\alpha_{1}^{(2)},G^{(2)}$ and $H^{(2)}$.
Moreover, from Lemma 2.6 and Theorem 2.7, we get the fact that, provided
$\\{w_{n}\\}$ is any solution of equation (2.9), we have
$\alpha_{0}^{(s)}=\alpha_{1}^{(s)}$ for all odd $s\geq 3$. It is easy to
derive the following recurrence relation:

$\displaystyle G_{1}^{(s+2)}$ $\displaystyle=$ $\displaystyle
G_{1}^{(s+1)}H^{(s+1)}+\frac{G^{(s+1)}}{G^{(s)}}\left(G_{1}^{(s+1)}-G_{1}^{(s)}H^{(s+1)}\right)$
(2.20)
$\displaystyle-\left(\frac{G^{(s+1)}}{G^{(s)}}\right)^{2}G^{(s)}H^{(s)}\;\;s\geq
2.$

As a consequence of Lemma 2.6 and Theorem 2.7, we get the following.

######  Theorem 2.8.

  * 1)

Any solution $\\{y_{n}\\}$ to equation (1.4) is also a solution to equation
(1.1) for $s=3$ with $H^{(3)}=H^{(2)}$. The first integrals, in this case, are
expressed via $G^{(2)},\;\;G_{1}^{(2)}$ and $H^{(2)}$ as

$G^{(3)}=G_{1}^{(2)}\;\;\mbox{and}\;\;G_{1}^{(3)}=G_{1}^{(2)}H^{(2)}+\left(G^{(2)}\right)^{2},$

where

$G^{(2)}=y_{1}\left(y_{0}+y_{1}+y_{2}-H^{(2)}\right)$

and

$G_{1}^{(2)}=y_{0}y_{1}y_{2}+y_{1}^{2}\left(y_{0}+y_{1}+y_{2}-H^{(2)}\right);$

  * 2)

any solution $\\{y_{n}\\}$ to equation (1.4) is also a solution to equation
(1.1) for all $s\geq 4$, where $H^{(s)}$ is defined by recurrent relation
(2.19) supplemented by (2.20) and the second relation in (2.18).

It is evident that, under the condition of Theorem 2.8, $H^{(s)}$, for all
$s\geq 4$, are unambiguously expressed as a rational function of variables
$(y_{0},y_{1},y_{2},H^{(2)})$.

######  Theorem 2.9.

Any solution $\\{f_{n}\\}$ to equation (2.14) is also a solution to equation
(2.13) for all $s\geq 3$, where the coefficients $\alpha_{n}^{(s)}$ and
$G^{(s)}$ are calculated via (2.18).

Proof. We prove this theorem by a contradiction method. To this aim, we
assume, that $\\{f_{n}\\}$ is a solution to (2.14) but at the same time it is
not a solution to $s$-th equation (2.13) for any fixed $s\geq 3$. Due to our
assumption $\\{w_{n}=f_{n}f_{n+2}/f_{n+1}^{2}\\}$ is a solution to equation
(2.9), and by Theorem 2.7, it is also a solution to $s$-th equation (2.8). But
if, according to our assumption, $\\{f_{n}\\}$ is not a solution to $s$-th
equation (2.13), then $\\{w_{n}\\}$ can not be a solution to $s$-th equation
(2.8). Thus, our assumption leads to a contradiction. $\Box$

##  3\. Construction of solution for difference equations

In this section, we give an example of a solution for equation (2.9). To this
aim, let us define the sequence $\\{T_{n}:n\geq 0\\}$ by linear recurrent
relation (1.5) with arbitrary initial condition, where $P_{0},P_{1}$ and $Q$
are supposed to be arbitrary numbers. It is obvious that $T_{n}$ can be
written as a linear combination $T_{n}=T_{0}N_{n}+T_{1}D_{n}$, where
$D_{n}=D_{n}(P_{0},P_{1},Q)$ and $N_{n}=N_{n}(P_{0},P_{1},Q)$ represent
particular solutions of (1.5) with initial conditions $(0,1)$ and $(1,0)$,
respectively. For example,

$D_{0}=0,\;\;D_{1}=1,\;\;D_{2}=P_{0},\;\;D_{3}=P_{0}P_{1}+Q,\;\;D_{4}=P_{0}\left(P_{0}P_{1}+2Q\right),\ldots$

and

$N_{0}=1,\;\;N_{1}=0,\;\;N_{2}=Q,\;\;N_{3}=QP_{1},\;\;N_{4}=Q\left(P_{0}P_{1}+Q\right),\ldots$

To clarify the meaning of these polynomials, for any $P_{0},P_{1}$ and $Q$,
let us consider the following 2-periodic “continued fraction”:

$C=\frac{Q}{P_{0}+\displaystyle{\frac{Q}{P_{1}+\displaystyle{\frac{Q}{P_{0}+\displaystyle{\frac{Q}{P_{1}+\cdots}}}}}}}$
(3.1)

and its convergents

$C_{2}=\frac{Q}{P_{0}},\;\;C_{3}=\frac{Q}{P_{0}+\displaystyle\frac{Q}{P_{1}}},\;\;C_{4}=\frac{Q}{P_{0}+\displaystyle\frac{Q}{P_{1}+\displaystyle\frac{Q}{P_{0}}}},\ldots$

Then, as is known, $C_{n}=N_{n}/D_{n}\;\;\forall n\geq 2$. Of course, $C$ is
literally a continued fraction only if $P_{0},P_{1}$ and $Q$ are integers.

######  Remark 3.1.

For the polynomials $D_{n}(P_{0},P_{1},Q)$ and $N_{n}(P_{0},P_{1},Q)$, one
knows the following (see, for example, [1]). The polynomial $D_{2n}/P_{0}$ and
$D_{2n+1}$, for any $n\geq 1$, are in fact homogeneous polynomial in
$P_{0}P_{1}$ and $Q$. Moreover, the sequence $\\{D_{n}:n\geq 0\\}$ is a
divisibility sequence. This fact can be illustrated by the formula

$D_{n}=\prod_{d|n}\Phi_{d}(P_{0}P_{1},Q),$

where, for example,

$\Phi_{1}=1,\;\;\Phi_{2}=P_{0},\;\;\Phi_{3}=P_{0}P_{1}+Q,\;\;\Phi_{4}=P_{0}P_{1}+2Q,$
$\Phi_{5}=P_{0}^{2}P_{1}^{2}+3P_{0}P_{1}Q+Q^{2},\;\;\Phi_{6}=P_{0}P_{1}+3Q,\ldots$

Let us define the sequence of polynomials $U_{n}=U_{n}(L,M)$ by recurrence
relations

$U_{2n}=U_{2n-1}-MU_{2n-2}\;\;\mbox{and}\;\;U_{2n+1}=LU_{2n}-MU_{2n-1}\;\;n\geq
1$ (3.2)

with $U_{0}=0$ and $U_{1}=1$. For example,

$U_{2}=1,\;\;U_{3}=L-M,\;\;U_{4}=L-2M,\;\;U_{5}=L^{2}-3LM+M^{2},\ldots$

It is known [1] that

$D_{n}(P_{0},P_{1},Q)=\left\\{\begin{array}[]{l}U_{n}(P_{0}P_{1},-Q),\;\;\mbox{if}\;\;n\;\;\mbox{odd}\\\\[2.84544pt]
P_{0}U_{n}(P_{0}P_{1},-Q),\;\;\mbox{if}\;\;n\;\;\mbox{even}\end{array}\right.$

and

$N_{n+1}(P_{0},P_{1},Q)=\left\\{\begin{array}[]{l}QU_{n}(P_{0}P_{1},-Q),\;\;\mbox{if}\;\;n\;\;\mbox{odd}\\\\[2.84544pt]
QP_{1}U_{n}(P_{0}P_{1},-Q),\;\;\mbox{if}\;\;n\;\;\mbox{even},\end{array}\right.$

and consequently $N_{n+1}(P_{0},P_{1},Q)=QD_{n}(P_{1},P_{0},Q)$.

To prepare the ground to prove Theorem 3.6 below, let us formulate some
lemmas.

######  Lemma 3.2.

By (1.5), the polynomials $T_{n}$ satisfy the identity

$T_{2n+2s}=D_{2s+2}T_{2n-1}+QD_{2s+1}T_{2n-2}$ (3.3)

and

$T_{2n+2s+1}=\frac{P_{1}}{P_{0}}D_{2s+2}T_{2n}+QD_{2s+1}T_{2n-1}$ (3.4)

for all $s\geq 0$.

Proof. Let us prove identity (3.3) by induction on $s$. To this aim, let us
suppose that relations (3.3) and (3.4) is valid for some $s\geq 0$. By the
first relation in (1.5), we have

$\displaystyle T_{2n+2s+2}$ $\displaystyle=$ $\displaystyle
P_{0}T_{2n+2s+1}+QT_{2n+2s}$ $\displaystyle=$ $\displaystyle
P_{0}\left(\frac{P_{1}}{P_{0}}D_{2s+2}T_{2n}+QD_{2s+1}T_{2n-1}\right)$
$\displaystyle+Q\left(D_{2s+2}T_{2n-1}+QD_{2s+1}T_{2n-2}\right)$
$\displaystyle=$ $\displaystyle
P_{1}D_{2s+2}T_{2n}+Q\left(P_{0}D_{2s+1}+D_{2s+2}\right)T_{2n-1}+Q^{2}D_{2s+1}T_{2n-2}$
$\displaystyle=$ $\displaystyle
P_{1}D_{2s+2}\left(QT_{2n-2}+P_{0}T_{2n-1}\right)+Q\left(P_{0}D_{2s+1}+D_{2s+2}\right)T_{2n-1}$
$\displaystyle+Q^{2}D_{2s+1}T_{2n-2}$ $\displaystyle=$
$\displaystyle\left(P_{0}\left(P_{1}D_{2s+2}+QD_{2s+1}\right)+QD_{2s+2}\right)T_{2n-1}$
$\displaystyle+Q\left(P_{1}D_{2s+2}+QD_{2s+1}\right)T_{2n-2}$ $\displaystyle=$
$\displaystyle D_{2s+4}T_{2n-1}+QD_{2s+3}T_{2n-2}.$

Thus, we have proved that if relations (3.3) and (3.4) are valid for some
$s\geq 0$, then (3.3) is also valid for $s+1$. Relations (3.3) and (3.4), for
$s=0$, are evidently valid by the condition of the lemma. Thus, by induction
on $s$, we prove this lemma. Identity (3.4) can be proved analogously. $\Box$

We actually need the following consequence of this lemma. Since $\\{D_{n}\\}$
represents a particular solution of linear equations (1.5), then from (3.3),
as a particular case, we derive the following relation:

$D_{2n+2s}=D_{2s+2}D_{2n-1}+QD_{2s+1}D_{2n-2}$ (3.5)

for all $s\geq 0$ and $n\geq 1$. As a consequence of (3.5) we get the
following lemma.

######  Lemma 3.3.

The polynomials $D_{r}(P_{0},P_{1},Q)$ satisfy the identity

$D_{r}\left(D_{r+3}-QD_{r+1}\right)=D_{r+2}\left(D_{r+1}-QD_{r-1}\right)\;\forall
r\geq 1.$ (3.6)

Proof. Indeed, assuming $n=k+2,s=k-1$ and $n=k+1,s=k$ in (3.5), we derive the
identities

$\displaystyle D_{4k+2}$ $\displaystyle=$ $\displaystyle
D_{2k}D_{2k+3}+QD_{2k-1}D_{2k+2}$ $\displaystyle=$ $\displaystyle
D_{2k+1}D_{2k+2}+QD_{2k}D_{2k+1}$

that gives (3.6) for $r=2k$. In turn, assuming $n=s=k+1$ and $n=k+2,s=k$ with
some $k\geq 1$ in (3.5), we derive the identities

$\displaystyle D_{4k+4}$ $\displaystyle=$ $\displaystyle
D_{2k+1}D_{2k+4}+QD_{2k}D_{2k+3}$ $\displaystyle=$ $\displaystyle
D_{2k+2}D_{2k+3}+QD_{2k+1}D_{2k+2}$

that gives (3.6) for $r=2k+1$. $\Box$

The following lemma plays crucial role for constructing some class of
solutions for difference equation (2.8).

######  Lemma 3.4.

A sequence $\\{T_{n}:n\geq 0\\}$ determined by the recursion (1.5) is also a
solution of bilinear equation (2.14), provided that

$\alpha^{(2)}_{n}=-\frac{P_{n}^{2}}{Q}\;\;\mbox{and}\;\;G^{(2)}=\frac{P_{0}P_{1}+Q}{Q}.$
(3.7)

Moreover, we have

$H^{(2)}=\frac{-P_{0}P_{1}+2Q}{Q}.$ (3.8)

Proof. This lemma can be proved by direct but quite cumbersome calculations.
$\Box$

It should be noted that the parameters $\alpha^{(2)}_{0},\alpha^{(2)}_{1}$ and
$G^{(2)}$ in (3.7) do not depend on $T_{0}$ and $T_{1}$.

As a consequence of Lemma 3.4, we derive the following two identities:

$P_{0}^{2}D_{s}D_{s+2}=\left(P_{0}P_{1}+Q\right)D_{s+1}^{2}-QD_{s-1}D_{s+3}\;\;\forall\;\mbox{odd}\;\;s\geq
1$ (3.9)

and

$P_{1}^{2}D_{s}D_{s+2}=\left(P_{0}P_{1}+Q\right)D_{s+1}^{2}-QD_{s-1}D_{s+3}\;\;\forall\;\mbox{even}\;\;s\geq
2.$ (3.10)

######  Lemma 3.5.

The relations

$\displaystyle
QD_{s}D_{s+1}\left(D_{s+1}D_{s+4}-D_{s+2}D_{s+3}\right)+D_{s+2}D_{s+3}\left(D_{s}D_{s+3}-D_{s+1}D_{s+2}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+P_{0}^{2}D_{s+1}^{3}D_{s+3}-P_{1}^{2}D_{s}D_{s+2}^{3}=0\;\;\forall\;\mbox{even}\;s\geq
2$ (3.11)

and

$\displaystyle
QD_{s}D_{s+1}\left(D_{s+1}D_{s+4}-D_{s+2}D_{s+3}\right)+D_{s+2}D_{s+3}\left(D_{s}D_{s+3}-D_{s+1}D_{s+2}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+P_{1}^{2}D_{s+1}^{3}D_{s+3}-P_{0}^{2}D_{s}D_{s+2}^{3}=0\;\;\forall\;\mbox{odd}\;s\geq
1$ (3.12)

are identities.

Proof. Let us prove only (3.12). By (3.9), for all odd $s\geq 1$, we have

$\displaystyle P_{1}^{2}D_{s+1}^{3}D_{s+3}-P_{0}^{2}D_{s}D_{s+2}^{3}$
$\displaystyle=$ $\displaystyle
D_{s+1}^{2}\left\\{(P_{0}P_{1}+Q)D_{s+2}^{2}-QD_{s}D_{s+4}\right\\}$
$\displaystyle-
D_{s+2}^{2}\left\\{(P_{0}P_{1}+Q)D_{s+1}^{2}-QD_{s-1}D_{s+3}\right\\}$
$\displaystyle=$ $\displaystyle
Q\left(D_{s-1}D_{s+2}^{2}D_{s+3}-D_{s}D_{s+1}^{2}D_{s+4}\right).$

Taking the latter into account, we can rewrite (3.12) as

$\displaystyle
QD_{s}D_{s+1}\left(D_{s+1}D_{s+4}-D_{s+2}D_{s+3}\right)+D_{s+2}D_{s+3}\left(D_{s}D_{s+3}-D_{s+1}D_{s+2}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;+Q\left(D_{s-1}D_{s+2}^{2}D_{s+3}-D_{s}D_{s+1}^{2}D_{s+4}\right)=0$

or quite simply as $QJ_{s-1}+J_{s}=0$, where, by definition,
$J_{s}=D_{s}D_{s+3}-D_{s+1}D_{s+2}$. But the latter is nothing but identity
(3.6), which has already been proven. Identity (3.11) can be proved using
similar reasoning. $\Box$

It follows, from Lemma 3.4, that the sequence
$\\{w_{n}=T_{n}T_{n+2}/T_{n+1}^{2}:n\geq 0\\}$ represents a particular
solution of equation (2.9). In turn, by Theorem 2.7, we know that this
sequence is also a solution of equation (2.8) for all $s\geq 3$. More exactly,
we have the following theorem.

######  Theorem 3.6.

The sequence $\\{w_{n}=T_{n}T_{n+2}/T_{n+1}^{2}:n\geq 0\\}$ is a particular
solution of equation (2.8), for any $s\geq 2$, provided that the parameters
$\alpha_{0}^{(s)},\alpha_{1}^{(s)}$ and $G^{(s)}$ are given by

$\alpha_{0}^{(s)}=-\frac{P_{0}D_{s}}{QD_{s-1}},\;\;\alpha_{1}^{(s)}=\left\\{\begin{array}[]{l}\alpha_{0}^{(s)},\;\;\mbox{if}\;$s$\;\mbox{odd}\\\\[5.69046pt]
\displaystyle\frac{P_{1}^{2}}{P_{0}^{2}}\alpha_{0}^{(s)},\;\;\mbox{if}\;$s$\;\mbox{even}\end{array}\right.\;\;\mbox{and}\;\;G^{(s)}=\frac{D_{s+1}}{QD_{s-1}}.$
(3.13)

In addition we have

$H^{(s)}=-G^{(s)}+s+1.$ (3.14)

Proof. To begin with, we note that formulas (3.13) and (3.14), in the case
$s=2$, give what we have in (3.7) and (3.8). Moreover, by (2.17), we get

$\alpha_{0}^{(3)}=\alpha_{1}^{(3)}=-\frac{P_{0}P_{1}+Q}{Q}\;\;\mbox{and}\;\;G^{(3)}=\frac{P_{0}P_{1}+2Q}{Q}$

and this corresponds to (3.13) for $s=3$. Also, one can check that the
equality $H^{(3)}=H^{(2)}$ is in accordance to (3.14). It remains to check
that the substitution of (3.13) and (3.14) into recurrent relations (2.18) and
(2.19) in Theorem 2.7 gives identities.

It is easy to check that substituting $\alpha_{0}^{(s)}$ and
$\alpha_{1}^{(s)}$ given by (3.13) into (2.18) gives trivial identities. In
turn, substituting (3.14) into (2.19) yields the relation

$G^{(s+1)}=G^{(s)}-\frac{G^{(s)}}{G^{(s-1)}}+1.$

Expressing here $G^{(s)}$ via $D_{s}$ as in (3.13), we get the identity (3.6)
that has already been proven. So it remains to check the fulfillment of the
second recurrent relation in (2.18). One can make sure that substituting
(3.13) and (3.14) into this relation gives identities (3.11) and (3.12) that
has already proven. $\Box$

In turn, Theorem 3.6 implies the following theorem.

######  Theorem 3.7.

The sequence $\\{y_{n}=T_{n}T_{n+3}/(T_{n+1}T_{n+2}):n\geq 0\\}$ is a solution
of the equation (1.1) with parameter $H^{(s)}$ given by (3.14) for all $s\geq
2$.

The following theorem characterizes the solution provided by the Theorem 3.7.

######  Theorem 3.8.

The sequence $\\{y_{n}=T_{n}T_{n+3}/(T_{n+1}T_{n+2}):n\geq 0\\}$ satisfies the
relation

$y_{n}+y_{n+1}+y_{n+2}-y_{n}y_{n+2}+\frac{1}{y_{n+1}}=3.$ (3.15)

Proof. We have

$G_{1}^{(2)}=G^{(3)}=\frac{P_{0}P_{1}+2Q}{Q}=y_{n}y_{n+1}y_{n+2}+y_{n+1}^{2}\left(y_{n}+y_{n+1}+y_{n+2}-H^{(2)}\right).$
(3.16)

By the relation

$G^{(2)}=\frac{P_{0}P_{1}+Q}{Q}=y_{n+1}\left(y_{n}+y_{n+1}+y_{n+2}-H^{(2)}\right),$
(3.17)

we can rewrite (3.16) as

$\frac{P_{0}P_{1}+2Q}{Q}=y_{n}y_{n+1}y_{n+2}+y_{n+1}\frac{P_{0}P_{1}+Q}{Q}.$

In turn we can rewrite this relation as

$y_{n+1}\left(y_{n}y_{n+2}-H^{(2)}-\frac{1}{y_{n+1}}+3\right)=\frac{P_{0}P_{1}+Q}{Q}=G^{(2)}$

Finally comparing the latter with (3.17), we get (3.15). $\Box$

Of course, as a particular case we get the following relation:

$y_{0}+y_{1}+y_{2}-y_{0}y_{2}+\frac{1}{y_{1}}=3$

that allows to parametrize the solution in terms of initial values.

At the end of this section, we would like to give examples of the sequence
$\\{T_{n}:n\geq 0\\}$. Let, for example, $P_{0}=P_{1}=Q=1$. Then recurrent
relations (1.5) is reduced to the relation $T_{n+2}=T_{n+1}+T_{n}$, that
defines generalized Fibonacci numbers. In particular, if $T_{0}=0$ and
$T_{1}=1$, then $T_{n}=D_{n}=F_{n}$, where $F_{n}$’s are the Fibonacci
numbers. In turn, if $T_{0}=2$ and $T_{1}=1$, then $T_{n}=2N_{n}+D_{n}=L_{n}$
, where $L_{n}$’s are Lucas numbers. Evidently, that $L_{n}=F_{n}+2F_{n-1}$.

Continued fraction (3.1), in this case, becomes

$C=\Phi-1=\frac{1}{1+\displaystyle{\frac{1}{1+\displaystyle{\frac{1}{1+\cdots}}}}},$

where ${\displaystyle\Phi={\frac{{\sqrt{5}}+1}{2}}}$ is well known as a golden
section. The sequence of convergents of $C$, in this case, are expressed via
the Fibonacci numbers as $C_{n}=F_{n-1}/F_{n}$.

More generally, let $P_{0}=P_{1}=P$. Let $P$ and $Q$ be relatively prime
integers such that discriminant $D=P^{2}+4Q\neq 0$ of the characteristic
equation $x^{2}-Px-Q=0$ is not zero. Let $a$ and $b$ be algebraic numbers such
that $a+b=P$ and $ab=-Q$. Finally, if a ratio $a/b$ is not a root of unity
then a Lucas sequence and its companion are defined as [11]

$D_{n}=\frac{a^{n}-b^{n}}{a-b}\;\;\mbox{and}\;\;2N_{n}+PD_{n}=a^{n}+b^{n},$

respectively.

Also as a special case, the sequence $\\{T_{n}:n\geq 0\\}$ contains a Lehmer
sequence, which, to a certain extent, has already been mentioned above. Let
$P_{0}=1$. Let us denote $P_{1}=L$ and $Q=-M$. Then linear recurrent relations
(1.5) reduce to relations (3.2). Let $L$ and $M$ be relatively prime integers
such that $D=L-4M\neq 0$, while $a$ and $b$ be algebraic numbers that are
roots of the characteristic equation $x^{2}-\sqrt{L}x+M=0$, that is,
$L=(a+b)^{2}$ and $M=ab$. In addition, let us suppose that the quotient $a/b$
is not a root of unity, then the Lehmer sequence is defined as [11]

$D_{n}=\left\\{\begin{array}[]{c}\displaystyle\frac{a^{n}-b^{n}}{a-b},\;\;\mbox{if}\;n\;\mbox{odd}\\\\[8.5359pt]
\displaystyle\frac{a^{n}-b^{n}}{a^{2}-b^{2}},\;\;\mbox{if}\;n\;\mbox{even}.\end{array}\right.$

To define a companion Lehmer sequence one needs to put $P_{0}=L$, $P_{1}=1$
and $Q=-M$. Then

$2N_{n}+D_{n}=\left\\{\begin{array}[]{c}\displaystyle\frac{a^{n}+b^{n}}{a+b},\;\;\mbox{if}\;n\;\mbox{odd}\\\\[8.5359pt]
\displaystyle a^{n}+b^{n},\;\;\mbox{if}\;n\;\mbox{even}.\end{array}\right.$

##  4\. Final remarks

Let us briefly discuss the results presented in this paper. In the end, we
find the relationship of integer sequences with integrable equations (1.1).
Such relationship can later find its applications in the theory of integer
sequences. It would also be interesting to explore these results for
investigation Volterra lattice integrable hierarchy (1.2).

Difference equations (1.1) represent, in fact, a subclass of two-parametric
class of equations [14]

$y_{n+k}\tilde{S}^{k}_{s+1}(n)=y_{n+s}\tilde{S}^{k}_{s+1}(n+1)\;\;k\geq
1,\;s\geq k+1.$ (4.1)

A precise description of these equations is as follows. Discrete polynomial
$\tilde{S}^{k}_{s}(n)$ in (4.1) can be written as

$\tilde{S}^{k}_{s}(n)=\sum_{j=0}^{k}(-1)^{j}H_{j}^{(k,s)}S^{k-j}_{s-j}(n+j),$

where $H_{j}^{(k,s)}$ are some parameters and $H_{0}^{(k,s)}=1$, while
discrete polynomials $S^{k}_{s}(n)$ are defined by (1.3). Evidently the
equations (1.1) can be rewritten as

$y_{n+1}\tilde{S}^{1}_{s+1}(n)=y_{n+s}\tilde{S}^{1}_{s+1}(n+1),\;s\geq 2.$

Actual calculations suggest that Theorem 2.8 can be expanded to this class of
equations. More exactly, we suppose that any solution to equation (1.4) is at
the same time a solution to equation (4.1), for all $k\geq 1$ and $s\geq k+1$,
provided that we restrict these parameters appropriately. Moreover we suppose
that this constraint looks like that

$H_{k}^{(k,s)}=\sum_{j=1}^{k-1}a_{j}^{(k,s)}H_{j}^{(k,s)}+b^{(k,s)},$

where the coefficients $a_{j}^{(k,s)}$ and $b^{(k,s)}$ are some rational
functions of variables $(y_{0},y_{1},y_{2},H^{(2)})$. Now we do not see a
direct way to make such a generalization, but, in the future, we intend to
continue our investigation in this direction.

## Appendix A

Let us prove here Theorem 2.7, by induction. For any $s\geq 4$, by (2.8), we
have

$\displaystyle\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}\cdot\prod_{j=0}^{s-1}w_{n+j+1}w_{n+j+2}$
$\displaystyle\;\;\;=\left(\alpha_{n}^{(s)}\prod_{j=1}^{s-1}w_{n+j}+G^{(s)}\right)\left(\alpha_{n+1}^{(s)}\prod_{j=1}^{s-1}w_{n+j+1}+G^{(s)}\right)$
$\displaystyle\;\;\;=\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}+G^{(s)}\prod_{j=2}^{s-1}w_{n+j}\left(\alpha^{(s)}_{n}w_{n+1}+\alpha^{(s)}_{n+1}w_{n+s}\right)$
$\displaystyle\;\;\;\;\;\;+\left(G^{(s)}\right)^{2}.$

We can rewrite (2.11) as

$\displaystyle\alpha^{(s)}_{n}w_{n+1}+\alpha^{(s)}_{n+1}w_{n+s}$
$\displaystyle=$ $\displaystyle
H^{(s)}\prod_{j=1}^{s}w_{n+j}-\frac{G^{(s)}}{\prod_{j=2}^{s-1}w_{n+j}}$
$\displaystyle-\prod_{j=1}^{s}w_{n+j}\sum_{j=1}^{s-1}w_{n+j}w_{n+j+1}.$

Taking the latter into account, we get

$\displaystyle\prod_{j=0}^{s-1}w_{n+j}w_{n+j+1}\cdot\prod_{j=0}^{s-1}w_{n+j+1}w_{n+j+2}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;=\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}+G^{(s)}\prod_{j=2}^{s-1}w_{n+j}\left(H^{(s)}\prod_{j=1}^{s}w_{n+j}\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left.-\frac{G^{(s)}}{\prod_{j=2}^{s-1}w_{n+j}}-\prod_{j=1}^{s}w_{n+j}\sum_{j=1}^{s-1}w_{n+j}w_{n+j+1}\right)+\left(G^{(s)}\right)^{2}.$

From this follows the fact that if $\\{w_{n}\\}$ satisfies equation (2.8), for
some $s\geq 4$, then it satisfies also the equation

$\prod_{j=0}^{s}w_{n+j}w_{n+j+1}=\left(\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}+G^{(s)}H^{(s)}\right)-G^{(s)}\sum_{j=1}^{s-1}w_{n+j}w_{n+j+1}.$
(A.1)

Let us now assume that $\\{w_{n}\\}$ is also a solution to $(s-1)$-th equation
(2.8). From (2.8) and (2.10), we derive the following relation:

$\sum_{j=1}^{s-1}w_{n+j}w_{n+j+1}=H^{(s-1)}+\frac{\alpha_{n}^{(s-1)}}{G^{(s-1)}}\left(\alpha_{n+1}^{(s-1)}-\prod_{j=1}^{s}w_{n+j}\right).$
(A.2)

Substituting (A.2) into (A.1), yields

$\displaystyle\prod_{j=0}^{s}w_{n+j}w_{n+j+1}$ $\displaystyle=$
$\displaystyle\left(\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}+G^{(s)}H^{(s)}\right)$
$\displaystyle-G^{(s)}\left(H^{(s-1)}+\frac{\alpha_{n}^{(s-1)}}{G^{(s-1)}}\left(\alpha_{n+1}^{(s-1)}-\prod_{j=1}^{s}w_{n+j}\right)\right)$
$\displaystyle=$
$\displaystyle\alpha_{n}^{(s+1)}\prod_{j=1}^{s}w_{n+j}+G^{(s+1)},$

where

$\alpha_{n}^{(s+1)}=\frac{G^{(s)}}{G^{(s-1)}}\alpha_{n}^{(s-1)}$ (A.3)

and

$G^{(s+1)}=\alpha_{n}^{(s)}\alpha_{n+1}^{(s)}+G^{(s)}H^{(s)}-\frac{G^{(s)}}{G^{(s-1)}}\left(\alpha_{n}^{(s-1)}\alpha_{n+1}^{(s-1)}+G^{(s-1)}H^{(s-1)}\right).$

In a result, we get the following fact. If $\\{w_{n}\\}$ is a solution to
$(s-1)$-th and $s$-th equation (2.8) simultaneously, then it is a solution to
$(s+1)$-th equation. On the other hand, from lemma 2.6 we know that if
$\\{w_{n}\\}$ is a solution of equation (2.9) then it is also a solution of
equations (2.16). It makes a base of induction. Thus, we are in a position to
prove the main part of Theorem 2.7 by induction on $s$.

Let us now prove relation (2.19). Further we use already proven fact that
under the condition of Theorem 2.7, the sequence $\\{w_{n}\\}$ is a solution
to the equation (2.8), for any $s\geq 3$, with corresponding coefficients.
Then, we have

$H^{(s+1)}-H^{(s)}=w_{n+s}w_{n+s+1}+\frac{\alpha_{n+1}^{(s+1)}}{\prod_{j=1}^{s}w_{n+j}}-\frac{\alpha_{n+1}^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}.$

By (2.8), we can write

$w_{n+s}w_{n+s+1}=\frac{\alpha_{n+1}^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}}+\frac{G^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}}.$
(A.4)

Taking into account relations (A.3) and (A.4), we have

$H^{(s+1)}-H^{(s)}=\frac{G^{(s)}}{\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}}\left(1+\frac{\alpha_{n+1}^{(s-1)}}{G^{(s-1)}}\prod_{j=2}^{s-1}w_{n+j}\right).$

Again, by (2.8),

$1+\frac{\alpha_{n+1}^{(s-1)}}{G^{(s-1)}}\prod_{j=2}^{s-1}w_{n+j}=\frac{\prod_{j=1}^{s-1}w_{n+j}w_{n+j+1}}{G^{(s-1)}}$

and using the latter we get (2.19). $\Box$

## Acknowledgments

I wish to thank the referees for carefully reading the article and for remarks
which improved the presentation.

The results were obtained within the framework of the state assignment of the
Ministry of Education and Science of the Russian Federation on the project No.
121041300058-1.

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