A note on general solutions to a hyperbolic-cotangent class of systems of difference equations

Recently there has been some interest in difference equations and systems whose forms resemble some trigonometric formulas. One of the classes of such systems is the so-called hyperbolic-cotangent class of systems of difference equations. The corresponding two-dimensional class has two delays denoted by k and l. So far the class has been studied for the case k = l, and it was shown that it is practically solvable when max{k, l} ≤ 2. In this note we show practical solvability of the system in the case k = l, not only for small values of k and l, but for all k = l ∈N, which is the first result of such generality.


Introduction
Let, as usual, N denote the set of natural numbers, N 0 of nonnegative integers, Z of integers, R of reals, C of complex numbers, and let k = l, m, where l, m ∈ Z, l ≤ m, denote the set of all integers k such that l ≤ k ≤ m.
Finding closed-form formulas for solutions to difference equations and systems is one of the first topics investigated in the branch of mathematics. One of the first solvable difference equations appearing in the literature was the following: a n = c 2 a n-1 + c 1 a n-2 , n ≥ 2, where c 1 , c 2 ∈ R, c 1 = 0 and c 2 2 = -4c 1 .
There are classes of difference equations which are similar to some trigonometric formulas. The similarity suggests potential solvability of the difference equations in the form. An example of such a difference equation appeared long time ago in [33]. One of such classes is the hyperbolic-cotangent class of difference equations (see, for instance, [15]).
Motivated by the hyperbolic-cotangent class of equations, by our paper [11], and in general by papers by Papaschinopoulos and Schinas on systems of difference equations (see, e.g., [20][21][22][23][24][25][34][35][36][37]), we have initiated an investigation of the corresponding classes of hyperbolic-cotangent systems of difference equations, that is, of the following ones: where k, l ∈ N 0 , parameter a and initial values are complex numbers, and where u n , v n , w n , and s n are x n or y n . We have shown that for some values of k and l, all the systems in (4) are solvable in closed form. More concretely, in [18] and [19] the systems when k = 0 and l = 1 were solved. Another solution to the solvability problem in this case was given in [13]. Further, in [12] the systems in the case when k = 1 and l = 2 were solved, and finally in [38] the systems were solved in the case k = 0 and l = 2. The methods used therein are closely related to those used in the study of product-type systems (see, for instance, [39] as well as the related references therein).
To complete a solution to the solvability problem for the systems in the case max{k, l} ≤ 2, it is needed to deal with the case when k = l. Hence, in this paper we study the following systems of difference equations: where k ∈ N 0 . Since for a = 0 some simple and obvious changes of variables transform the systems in (5) to some homogeneous linear ones with constant coefficients, the case will be omitted (see also [13]).
First note that if k ∈ N, then system (5) is with interlacing indices [14,16]. Namely, let for m ∈ N 0 , and i = 0, k. Then it is easy to see that for every m ∈ N 0 , and i = 0, k. Hence, the sequences (x (i) m , y (i) m ) m∈N 0 , i = 0, k, are solutions to system (5) with k = 0, implying that each solution to (5) consists of k + 1 unrelated solutions to the system in the case k = 0. This shows that it is of some interest to study only the systems in the case when k = 0, which is done in the sections that follow.

Product-type systems associated with the ones in (5)
The systems of difference equations in (5) with k = 0 are naturally connected to some product-type ones. To show this, first note that the following simple relations hold: for n ∈ N 0 . Now note that the system of difference equations (7) consists of the following nine ones: for n ∈ N 0 .
The following substitutions so that (8)-(16) become for n ∈ N 0 . Now note that if we show the solvability of systems (18)- (26), this together with the two relations in (17) will show the solvability of systems (8)- (16). Because of this, our main task is to show that there are closed-form formulas for solutions to systems (18)-(26).

Main results
This section considers the problem of solvability of systems (18)- (26). The systems are considered separately, one by one. It is shown that they all are really solvable by presenting some closed-form formulas for their general solutions. As a consequence, some closed-form formulas for general solutions to systems (8)- (16) are obtained. In this way it is shown that each of the systems of difference equations is practically solvable.

System (18)
From the first equation in the system of difference equations (18) we have from which by iteration and a simple inductive argument we obtain ζ n = ζ 2 n 0 , n ∈ N 0 .
By using (28) in the second equation in (18) it follows that Using relations (28) and (29) in (17), we see that the following theorem holds.
Theorem 1 Let a = 0. Then the following closed-form formulas present a general solution to system (8).

System (19)
Bearing in mind that the first equation in (19) is the same as in (18), we have that formula (28) also holds in this case. Employing (28) in the second equation in (19), we have From (30) Using relations (28) and (31) in (17), we see that the following theorem holds.
Theorem 2 Let a = 0. Then the following closed-form formulas present a general solution to system (9).

System (20)
Since the first equation in (20) is the same as in (18), formula (28) also holds in this case. On the other hand, the second equation in (20) is obtained from the first one by interchanging letters ζ and η, from which along with (28) it follows that Using relations (28) and (32) in (17), we see that the following theorem holds.
Theorem 3 Let a = 0. Then the following closed-form formulas present a general solution to system (10).
If we use the following notations: a 1 := 1 and b 1 := 2, then equation (33) can be written as follows: Employing relation (33), where index n is replaced by n -1 in (34), we have where a 2 and b 2 are clearly defined by a 2 := a 1 + b 1 and b 2 := 2a 1 .
Moreover, relation (40) can be used to calculate a n also for n ≤ 0 by using the following obvious consequence of it: a n-2 = a na n-1 2 .
By using (42) it is easy to see that formula (39) holds also for n = 0. The characteristic polynomial associated with equation (40) is P 2 (s) = s 2s -2, and clearly their roots are s 1 = 2 and s 2 = -1.
Using (44) in (39), we get By using (45) in the second equation in (21), we obtain Using relations (45) and (46) in (17), we see that the following theorem holds.
Theorem 4 Let a = 0. Then the following closed-form formulas present a general solution to system (11).

System (22)
First note that we have Using (47) in the first equation in (22), we have that (27) holds, but this time for n ≥ 2, from which it follows that and consequently, From (47) and (48) we have Using relations (48) and (49) in (17), we see that the following theorem holds.
Theorem 5 Let a = 0. Then the following closed-form formulas present a general solution to system (12).

System (23)
This system is obtained from the system in (19) by interchanging letters ζ and η. Hence, we have that the following formulas hold: and Using relations (50) and (51) in (17), we see that the following theorem holds.
Theorem 6 Let a = 0. Then the following closed-form formulas present a general solution to system (13).

System (24)
By using the second equation in (24) in the first one, we obtain From (52) we have ζ 2n = ζ 4 2(n-1) , n ∈ N, from which by iteration and a simple inductive argument we obtain and ζ 2n+1 = ζ 4 2(n-1)+1 , n ∈ N, from which by iteration and a simple inductive argument we obtain Since this system is symmetric, we have and Using relations (53)-(56) in (17), we see that the following theorem holds.
Theorem 7 Let a = 0. Then the following closed-form formulas present a general solution to system (14).

System (25)
This system is obtained from the system in (21) by interchanging letters ζ and η. Hence, we have that the following formulas hold: Using relations (57) and (58) in (17), we see that the following theorem holds. , n ∈ N 0 , present a general solution to system (15).

System (26)
This system is obtained from the system in (18) by interchanging letters ζ and η. Hence, we have that the following formulas hold: ζ n = η 2 n 0 , n ∈ N, and η n = η 2 n 0 , n ∈ N 0 .
Using (59) in (17), we see that the following theorem holds. , n ∈ N 0 , present a general solution to system (16).