Identities for linear recursive sequences of order $2$

1.   Introduction

Albert Girard published a class of identities in Amsterdam in 1629 and Edward Waring published similar material in Cambridge in 1762-1782, which are referred as Girard-Waring identities later. These identities may be derived from the earlier work of Sir Isaac Newton. Surveys and some applications of these identities can be found in Comtet [5] (P. 198), Gould [10], Shapiro and one of the authors [13], and the authors [15]. We now give a different approach to derive Girard-Waring identities by using the Binet formula of recursive sequences and divided differences. Meanwhile, this approach offers some formulas and identities that may have more wider applications. Finally, an application of the Girard-Waring identities to the sum of powers of consecutive integers is studied.

This paper starts from a review of the application of recursive sequences in the construction of a combinatorial identity called generalized Girard-Waring identity from the Binet formula and the generating function of a recursive sequence. By using the generalized Girard-Waring identity, the Binet type Girard-Waring identity is derived. Then the generalized Girard-Waring identity is used to develop several transformation formulas for recursive sequences of numbers and polynomials. All those results are shown in Nie, Chen, and the authors [16].

A recursive sequence constructed from a recursive relation starting from a few initial quantities models some real world problems or mathematical problems. As a natural mathematical model, recursive sequences are widely used in Combinatorics and Graph Theory, Number Theory, Fractal, Cryptography, etc. Many recursive number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations of various orders. Throughout this paper, a number sequence $\{ a_n\}$ is called a linear recursive sequence of order $2$ if it satisfies the following linear recurrence relation of order $2$:

for constants $p, q\in{ {\mathbb R}}$ and $q\not = 0$ and initial conditions $a_0$ and $a_1$. Let $\alpha$ and $\beta$ be two roots of the quadratic equation $x^2-p x-q = 0,$ of which the left-hand side is called the characteristic polynomial of the recurrence relation. From He and Shiue [14] (see also in [11,12,17] for some applications), the general term of the sequence $a_{n}$ can be presented by the following Binet formula.

In [16], the following expression of the general term of the recursive sequence defined by (1) is given.

Theorem 1.1. ([16]) Let $(a_n)$ be the sequence defined by the recursive relation (1), and let $\alpha$ and $\beta$ be two distinct roots of the characteristic polynomial of (1). Then we have the following generalized Girard-Waring identity:

Particularly, if $a_0 = 0$ and $a_1 = 1$, (3) implies the Binet type Girard-Waring identity

where $p = \alpha+\beta$ and $q = -\alpha\beta$.

As a source of Binet Girard-Waring identity, the generalized Girard-Waring identity (3) and its extension to recursive polynomial sequences have many applications including a simple way in transferring recursive sequences of numbers and polynomials. Recall that the Chebyshev polynomials of the first kind defined by the recurrence relation

for all $n\geq 2$ with initial conditions $T_0(x) = 1$ and $T_1(x) = x$, from Theorem 1.1 we have

Similarly, for Lucas numbers defined by

for all $n\geq 2$ and $L_0 = 2$ and $L_1 = 1$, we have

From the expressions of $T_n(x)$ and $L_n$, we may see that

or equivalently,

where $i = \sqrt{-1}$.

In general, [16] presents the following result for transferring a certain class of recursive sequences to the Chebyshev polynomial sequence of the first kind at certain points.

Theorem 1.2. ([16]) Let $\{ a_n\}_{n\geq 0}$ be a sequence defined by (1) with $pa_0 = 2a_1$, $a_0\not = 0$, and let $\{T_n(x)\}_{n\geq 0}$ be the Chebyshev polynomial sequence of the first kind defined by (6). Then

where

Namely, $a_n$ shown in (8) can be expressed as

Theorem 1.2 can be extended to recursive polynomial case as follows.

Corollary 1. ([16]) Let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x)$ and $a_1(x)$ satisfying $p(x)a_0(x) = 2a_1(x)$. Then

where $T_n(x)$ is the $n$th Chebyshev polynomial of the first kind.

The proofs of Theorem 1.2 and Corollary 2 can be found in [16]. In addition, [16] considers the recursive polynomial sequences defined by (1) with initial conditions $a_0(x) = 0$ and $a_1(x)\not = 0$, where $p = p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$. For instance,

for all $n\geq 1$, where initial conditions are $\hat U_0(x) = 0$ and $\hat U_1(x) = 1$. It is obvious that $\hat U_{n+1}(x) = U_n(x)$, the Chebyshev polynomials of the second kind. By using (3), we have

If $a_n$ is a sequence defined by (3), with the initial conditions $a_0 = 0$ and $a_1\not = 0$, direct substitution shows that

Comparing with the rightmost sides of (11) and (12), the following result is obtained.

Theorem 1.3. ([16]) Let $\{ a_n\}_{n\geq 0}$ be the sequence defined by (1) with $a_0 = 0$ and $a_1\not = 0$, and let $\{U_n(x)\}_{n\geq 0}$ be the Chebyshev polynomial sequence of the second kind defined by (10), where $U_n(x) = \hat U_{n+1}(x)$. Then

where

Namely,

Theorem 1.3 is extended in [16] to recursive polynomial case as Chebyshev polynomials of the second kind.

Corollary 2. Let $\{a_n(x)\}_{n\geq 0}$ be the recursive polynomial sequence defined by

where $p(x)\in{ {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x) = 0$ and $a_1(x)\not = 0$. Then

Numerous examples of the transformation of recursive number and polynomial sequences are shown in [16]. In the next section, we will see how those transformation help us to verify some well-known identities readily and to establish new identities.

2.   Identities constructed from the recursive sequence transformation

The Catalan identity for Fibonacci numbers $F_n$, namely that $F_n^2-F_{n+k}F_{n-k} = (-1)^{n-k}F_k^2$, and its special case of $k = 1$, named the Cassini identity for Fibonacci numbers $F_n$, namely that $F_{n+1}F_{n-1}-F_n^2 = (-1)^n$, are two facts about the Fibonacci numbers that one might call common mathematical knowledge (cf. [22,30,32]). We will in the following aim at presenting the Catalan-like and the Cassini-like identities in a more general context and in the process obtain similar results for related sequences by using the sequence transformation technique shown before.

From Theorem 1.3 we may get the Catalan-like identities of the recursive sequences defined by (1) as follows.

Theorem 2.1. Let $\{ a_n\}_{n\geq 0}$ be the sequence defined by (1) with $a_0 = 0$ and $a_1\not = 0$. Then, we have Catalan-like identity

Particularly, for $k = 1$, we have the Cassini-like identity for $(a_n)_{n\geq 0}$,

Proof. Let $\{U_n(x)\}_{n\geq 0}$ be the Chebyshev polynomial sequence of the second kind defined by (10). Then, from (13) we have

where $x_0 = \pm ip/(2\sqrt{q}).$ It is known (cf. Udrea [30]) that $U_n(x)$ satisfied the identity

Substituting (16) into (17) yields

which can be simplified to

Thus, we obtain (14). When $k = 1$, (14) implies (15).

Similarly, we have an analogue for the recursive polynomial sequence.

Corollary 3. Let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x) = 0$ and $a_1(x)\not = 0$. Then, we have Catalan-like identity

Particularly, for $k = 1$, we have the Cassini-like identity

Example 2.2. For the Fibonacci number sequence defined by $F_{n} = F_{n-1}+F_{n-2}$ $(n\geq 2)$, $F_0 = 0$, and $F_1 = 1$, we have $q = 1$ and the Catalan-like identity

and the Cassini-like identity

As for the Fibonacci polynomial sequence defined by $F_{n}(x) = xF_{n-1}(x)+F_{n-2}(x)$ $(n\geq 2)$, $F_0(x) = 0$, and $F_1(x) = 1$, we have $q = 1$ and the Catalan-like identity

and the Cassini-like identity

Example 2.3. For the Pell polynomial sequence defined by $P_n(x) = 2xP_{n-1}(x)+P_{n-2}(x)$ $(n\geq 2)$, $P_0(x) = 0$, and $P_1(x) = 1$, and the Pell number sequence $(P_n = P_n(1))_{n\geq 0}$, we have the Catalan-like identities

and the Cassini-like identities

Horadam and Mahon [22] prove the above identities by using a different approach.

Example 2.4. We call an integer $n\geq 2$ a balancing number (cf., for example, Behera and Panda [2]) if

for some $r\in{ {\mathbb N}}$. Here $r$ is called the balancer corresponding to the balancing number $n$. For example, $6, 35$, and $204$ are balancing numbers with balancers $2, 14$, and $84$, respectively.

It follows from (21) that, if $n$ is a balancing number with balancer $r$, then

and thus

Denote $(B_n)_{n\geq 0}$ the balancing number sequence and assume $B_0 = 0$. In [2] the recursive relation of the balancing number sequence is given as $B_{n} = 6B_{n-1}-B_{n-2}$ for $n\geq 2$ with the initials $B_0 = 0$ and $B_1 = 1$. Hence, we have the Catalan-like numbers and the Cassini-like numbers for the balancing number sequence as

The above two identities are given by Catarino, Campos, and Vasco [4] using a different approach.

The Chebyshev polynomials of the first kind $T_n(x)$ defined by (6) satisfies the Cassini-like identity (cf. [32])

Then, by using Theorem 1.2 we may find the analogy Cassini-like identities for some other recursive sequences. More precisely, we have the following result.

Theorem 2.5. Let $\{ a_n\}_{n\geq 0}$ be a sequence defined by (1) with $pa_0 = 2a_1$, $a_0\not = 0$. Then we have the Cassini-like identity for $(a_n)_{n\geq 0}$,

Similarly, let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x)$ and $a_1(x)$ satisfying $p(x)a_0(x) = 2a_1(x)$. Then

Proof. The transformation formula (8) shown in Theorem 1.2 with $x_0 = \pm ip/(2\sqrt{q})$ and $p = 2a_1/a_0$ gives

Substituting the above expression of $T_n(x_0)$ into (22) and noting $p = 2a_1/a_0$ yields

After simplifying the leftmost side of the last equation, we obtain the identity

which generalizes (23). Identity (24) can be proved similarly.

Remark 2.6. From the following examples, we will see many recursive sequences satisfy the condition $p = 2a_1/a_0$. It worth investigating the reason behind the fact.

Example 2.7. Consider the Lucas polynomial sequence defined by

with initials $L_0(x) = 2$ and $L_1(x) = x$ satisfying $p(x) = x = 2L_1(x)/L_0(x)$ and Lucas numbers $L_n = L_n(1)$. From (23) and (24) we have the Cassini-like identities

The Fermat polynomial sequence is defined by

with initials $f_0(x) = 2$ and $f_1(x) = x$ satisfying $p = x = 2f_1(x)/f_0(x)$, and the Fermat number sequence is defined by $(f_n = f_n(1))_{n\geq 0}$. Then, from (23) and (24) we get the Cassini-like identities

For the Dickson polynomial sequence of the first kind defined by

with initials $D_0(x, a) = 2$ and $D_1(x, a) = x$, it satisfies $p = x = 2D_1(x, a)/D_0(x, a)$. Then, from (23) we get the Cassini-like identity

which seems to be a new identity to the best of our knowledge.

For the Pell-Lucas polynomials A122075 [27] $Q_n(x)$ defined by (see Horadam and Mahon [22])

for all $n\geq 2$ with initial conditions $Q_0(x) = 2$ and $Q_1(x) = 2x$, we have the Cassini-like identity

For the Viate polynomials of the second kind defined by (see Horadan [21])

for all $n\geq 2$ with the initial conditions $v_0(x) = 2$ and $v_1(x) = x$, we have the Cassini-like identity

3.   An efficient and unified approach to construct product expansions and product expressions of recursive sequences

The transformation technique presented in the previous section provides an efficient and unified way to construct product expansions and product expressions for two classes recursive sequences from the corresponding product expansions and product expressions for the sequence of Chebyshev polynomials of the first kind and the second kind. The first class is a recursive number sequence $\{ a_n\}_{n\geq 0}$ defined by (1) with the initial conditions satisfying $pa_0 = 2a_1$ and $a_0\not = 0$ (the conditions of Theorem 1.2) or satisfying $a_0 = 0$ and $a_1\not = 0$ (the conditions of Theorem 1.3). The second class is a recursive polynomial sequence $\{ a_n(x)\}_{n\geq 0}$ with the initial conditions satisfying $p(x)a_0(x) = 2a_1(x)$ and $a_0(x)\not = 0$ (the conditions of Corollary 1) or satisfying $a_0(x) = 0$ and $a_1(x)\not = 0$ (the conditions of Corollary 2).

We first discuss the transformation of product expansions, then the transformation of product expressions. We know that the Chebyshev polynomials of the first kind satisfies the following product expansion

This follows quickly from the fact that $T_n(\cos \theta) = \cos n\theta$, $\theta = \arccos x$, and the addition theorem $2\cos \alpha \cos \beta = \cos(\alpha+\beta)+\cos (\alpha-\beta)$ with $\alpha = m\arccos x$ and $\beta = n\arccos x$. Consequently, we have the following result.

Theorem 3.1. Let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x)$ and $a_1(x)$ satisfying $p(x)a_0(x) = 2a_1(x)$. Then,

Proof. We have the following analogy of (9) for the Chebyshev polynomial sequence of the first kind

from which

Substituting the above expression for $T_n$ into (29) yields (30).

Example 3.2. For the Dickson polynomials shown in Example 2.7, from Theorem 3.1 and noting $a_0(x) = 2$ and $q = -a$, we have the identity

This identity seems new to our knowledge, although its special cases of $m = n$ and $m = n+1$, i.e.,

respectively, have been shown in Lidl, Mullen, and Turnwald, [23,p.11].

For the Pell-Lucas polynomials $Q_n(x)$ defined by (27), from Theorem 3.1 and noting $Q_0(x) = 2$ and $q = 1$, we have

The special cases of $m = n$ and $m = n+1$ are

respectively.

For the Viate polynomials of the second kind defined by (28), from Theorem 3.1 and noting $v_0(x) = 2$ and $q = -1$, we have the identity

The special cases of $m = n$ and $m = n+1$ are

respectively.

For the Lucas polynomial sequence $\{L_n(x)\}$ defined by (25), we have the identity

The special cases of $m = n$ and $m = n+1$ are

respectively.

Other examples for the Lucas number sequence are:

For Chebyshev polynomials of the second kind, it is known that their products can be written as

for $m\geq n$. By this with $n = 2$, there is a recurrence formula for Chebyshev polynomials of the second kind,

We now extend the above formula to the product formulas for a class of recursive sequences.

Theorem 3.3. Let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x) = 0$ and $a_1(x)\not = 0$. Then,

which is independent of $p(x)$. Particularly, for $n = 2$, we have the recursive formula

Proof. From Corollary 2, we have

where $x_0 = \pm ip(x)/(2\sqrt{q})$. Hence,

Substituting the above expression of $U_n(x)$ into (32) yields

The last equation can be simplified to

which implies (34). Formulas (35) follows after substituting $n = 2$.

Remark 3.4. Clearly, Theorem 3.3 can be extended to the recursive number sequence $\{ a_n\}_{n\geq 0}$ defined by

where $p, q\in { {\mathbb R}}$, with initial conditions $a_0 = 0$ and $a_1\not = 0$.

Example 3.5. For the Fibonacci polynomial sequence $\{F_n(x)\}$ defined in Example 2.2 with $p(x) = x$, $q = 1$, $F_0(x) = 0$ and $F_1(x) = 1$, from (34) and (35) in Theorem 3.3 we have product expansion and recursive formula for Fibonacci polynomials.

For Fibonacci numbers $F_n = F_n(1)$, we have

For the Pell polynomial sequence defined in Example 2.3 with $q = 1$ and $P_1(x) = 1$, we have their product expansion and recursive formula as follows.

For the Pell number sequence $(P_n = P_n(1))_{n\geq 0}$, we have their product expansion and recursive formula as

For the balancing numbers defined in Example 2.4 by $B_{n} = 6B_{n-1}-B_{n-2}$ for $n\geq 2$ with the initials $B_0 = 0$ and $B_1 = 1$, noting $q = -1$ we have the product expansion and recursive formula for $(B_n)_{n\geq 0}$ as

In Cahill, D'Errico, and Spence [3], the following product expressions for the Fibonacci number sequence and the Lucas number sequence were constructed:

for $n\geq 2$ and $n\geq 1$, respectively. In an earlier paper, Hoggatt and Bicknell [19] showed that the roots of Fibonacci polynomials $F_n(x)$ and Lucas polynomials $L_n(x)$ of degree $n$ are $x = 2i\cos (k\pi)/n$ $(k = 1, 2, \ldots, n-1)$ and $x = 2i\cos (2k-1)\pi/(2n)$ $(k = 1, 2, \ldots, n)$, respectively, which imply the identities

We will show that identities (36)-(39) can be easily established by using our sequence transformation technique. Actually, they are special cases of the general results of the following two theorems.

In the handbook edited by Zwillinger [33], the product expression of Chebyshev polynomials of the first kind and the second kind are given as

Theorem 3.6. Let $\{a_n\}_{n\geq 0}$ be a recursive number sequence defined by (1) with initial conditions $a_0$ and $a_1$ satisfying $pa_0 = 2a_1$. Then, we have

Similarly, let $\{ a_n(x)\}_{n\geq 0}$ be a recursive polynomial sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ $(p\in { {\mathbb R}})$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x)$ and $a_1(x)$ satisfying $p(x)a_0(x) = 2a_1(x)$. Then, we have

Proof. We now prove the formula (42) for the polynomial sequence, the corresponding result for the number sequence can be proved similarly. We have the following analogy of (9) for the Chebyshev polynomial sequence of the first kind

from this equation and (40) with the replacement $x\to \pm \frac{i p(x)}{2\sqrt{q}}$ we have

The equivalence of two product expression can be shown by setting $k\to n-k+1$. Indeed, we have

Example 3.7. For the Lucas polynomial sequence $\{L_n(x)\}$ defined by (25) with $p(x) = x$, $q = 1$, $a_0(x) = 2$ and $a_1(x) = x$ satisfying $p(x)a_0(x) = 2x = 2a_1(x)$, from Theorem 3.6 we immediately have (39). For Lucas numbers $L_n = L_n(1)$, we also readily obtain (37).

For the Dickson polynomials shown in Example 2.7, from Theorem 3.6 and noting $a_0(x) = 2$, $a_1(x) = D_1(x, a) = x$, and $q = -a$, we have the identity

Particularly, for $a = 1$

Furthermore, $D_n(x, a)$ has roots $2\sqrt{a}\cos \frac{(2k-1)\pi}{2n}$ for $k = 1, 2, \ldots, n$.

For the Pell-Lucas polynomials $Q_n(x)$ defined by (27) with $p(x) = 2x$, $q = 1$, $Q_0(x) = 2$ and $Q_1(x) = 2x$ satisfying $p(x)Q_0(x) = 4x = 2Q_1(x)$, from Theorem 3.6 we have the product expression of $Q_n(x)$ as

and $Q_n(x)$ has roots $i\cos \frac{(2k-1)\pi}{2n}$ for $k = 1, 2, \ldots, n$.

For the Viate polynomials of the second kind defined by (28) with $p(x) = x$, $q = -1$, $v_0(x) = 2$ and $v_1(x) = x$ satisfying $p(x) v_0(x) = 2x = 2v_1(x)$, from Theorem 3.6 we have the product expression of $v_n(x)$ as

and $v_n(x)$ has roots $2\cos \frac{(2k-1)\pi}{2n}$ for $k = 1, 2, \ldots, n$.

Theorem 3.8. Let $\{ a_n(x)\}_{n\geq 0}$ $(a_n))_{n\geq 0})$ be a recursive polynomial (number) sequence defined by

where $p(x)\in { {\mathbb R}}[x]$ $(p\in { {\mathbb R}})$ and $q\in{ {\mathbb R}}$, with initial conditions $a_0(x) = 0$ $(a_0 = 0)$ and $a_1(x)\not = 0$ $(a_1\not = 0)$. Then, we have

where the equivalence of the two expressions in (44) ((43)) can be seen from the transformation $k\to n-k+1$.

Proof. Let $\{U_n(x)\}_{n\geq 0}$ be the Chebyshev polynomial sequence of the second kind defined by (10). From Corollary 2 and (41) we have

where the equivalence of the last two products can be shown by setting $k\to n-k$ into one product as

Similarly, for recursive number sequence $(a_n)_{n\geq 0}$ we have

Example 3.9. For the Fibonacci polynomial sequence $\{F_n(x)\}$ defined in Example 2.2 with $p(x) = x$, $q = 1$, $F_0(x) = 0$ and $F_1(x) = 1$, from Theorem 3.8 we immediately have (38). For Fibonacci numbers $F_n = F_n(1)$, we also readily obtain (36).

For the Pell polynomial sequence defined in Example 2.3 by $P_n(x) = 2xP_{n-1}(x)+P_{n-2}(x)$ $(n\geq 2)$, $P_0(x) = 0$, and $P_1(x) = 1$, and the Pell number sequence $(P_n = P_n(1))_{n\geq 0}$, we have their product expressions as

Furthermore, $P_n(x)$ has roots $i \cos\frac{k\pi}{n}$ for $k = 1, 2, \ldots, n$.

For the balancing polynomials defined by the recurrence $B_n(x) = 6xB_{n-1}(x)-B_{n-2}(x)$ $(n\geq 2)$, $B_0(x) = 0$, and $B_1(x) = 1$ (cf. Frontczak [9]), we have have their product expressions as

which was also proved by Ray [28] using a different approach.

For the balancing numbers defined by $B_n = B_n(1)$ or by $B_{n} = 6B_{n-1}-B_{n-2}$ for $n\geq 2$ with the initials $B_0 = 0$ and $B_1 = 1$ (cf. Example 2.4), we have their product expression

This formula was proved in Ray [29] using a more complicated approach.

4.   Identities raised from a Diophantus problem

Diophantus raised the following problem (Heath [18,pp.179-181] and [7,p. 513], ): "To find four numbers such that the product of any two increased by unity is a square", for which he obtained the solution $1/16$, $33/16$, $68/16$, $105/16$.

Fermat [8] (cf. p. 251) found the solution $1, 3, 8,120$. In 1968, J.H. van Lint raised the problem whether the number $120$ is the unique (positive) integer $n$ for which the set $\{1, 3, 8, n\}$ constitutes a solution for the above Diophantus' problem; in the same year, Baker and Davenport [1] studied this question and concluded that, in fact, $120$ is the unique integer satisfying the problem raised by J.H. van Lint. In 1977, Hoggatt and Bergum [19] observed that $1, 3, 8$ are, respectively, the terms $F_2$, $F_4$, $F_6$ of the Fibonacci sequence and formulated the problem of finding a positive integer $n$ such that $F_{2t^n}+1, F_{2t+2^n}+1, F_{2t+4^n}+1$ be perfect squares. Hoggatt and Bergum also obtained the number $n = 4F_{2t+1}F_{2t+2}F_{2t+3}$, which, for $t = 1$, gives exactly $n = 120$.

The result shown in [19] was generalized in Morgado [25] by showing that the product of any two distinct elements of the set

increased by $\pm F^2_aF^2_b$ with suitable integers $a$ and $b$ is a perfect square. In other words, this set is a Fibonacci quadruple. This result was fourthly generalized in Horadam [20] by presenting that the product of any two distinct elements of the set

where $w_n = w_n(a, b;p, q) = pw_{n-1}-qw_{n-2}$, $w_0 = a$, and $w_1 = b$, increased by a suitable integer, is a perfect square. In other words, this set is a Diophantine quadruple, which is a generalization of Fibonacci quadruple because $F_n = w_n(0, 1;1, -1)$. Related work can be found in Melham and Shannon [24] and Cooper [6].

Udrea [31] generalizes a result obtained in [25] for the Fibonacci sequence and $(U_n = U_n(x))_{n\geq 0}$, the sequence of Chebyshev polynomials of the second kind, more precisely, the product of any two distinct elements of the set

increased by $U_a^2U_b^2$, for suitable nonnegative integer $a$ and $b$ is a perfect square.

Morgado [26] proved an analogue result for $(T_n = T_n(x))_{n\geq 0}$, the sequence of Chebyshev polynomials of the first kind. More precisely, the product of any two distinct elements of the set

increased by $((T_h-T_k)/2)^t$, where $T_h$ and $T_k$, with $k>h\geq 0$, are suitable terms of the sequence $(T_n)_{n\geq 0}$, is a perfect square.

We may use the sequence transformation technique to find the increased integers of the Diophantine quadruple (46) when $n = 2m$, $m\geq 0$, for some special set $\{a, b, p, q\}$.

Theorem 4.1. Let $\{ a_n\}_{n\geq 0}$ be a sequence defined by $a_{n} = pa_{n-1} +a_{n-2}$, $n\geq 2$, with $a_0 = 0$ and $a_1 = 1$. Then, we have a Diophantine quadruple

for $m, r\in{ {\mathbb N}}$. More precisely, the product of any two distinct elements of the set increased by $(a_\alpha a_\beta)^2$ for suitable $\alpha, \beta \in{ {\mathbb N}}$ is a perfect square.

Proof. From (2.1) of [30], the sequence of Chebyshev polynomials of the second kind satisfy

for any $x\in{ {\mathbb C}}$, and $m, r, s\in{ {\mathbb N}}$, and noting (16),

where $x_0 = \pm ip/(2\sqrt{q})$, we immediately have

In the last equation, canceling $(\pm i)^{2m+r+s}$ and then replacing $m+1$ by $2m$ yields

By setting $s = r$ in (50), one obtain

for $m, r\in { {\mathbb N}}$. Let us replace $2m$ by $2m+2r$ in (51). Then

for $m, r\in{ {\mathbb N}}$. Identities (51) and (52) prove the theorem partially with $\alpha = r$ and $\beta = 1$.

Let us replace $r$ by $2r$ in (51). Then,

for $m, r\in{ {\mathbb N}}$, which proves the theorem partially with $\alpha = 2r$ and $\beta = 0$.

From identity (50), it follows

and so

for $m, r, s\in{ {\mathbb N}}$. By setting $s = 2r$ into (54), one obtains

for $m, r\in{ {\mathbb N}}$. Let us replace $2m$ by $2m+r$ in (55). Then it becomes

for $m, r\in{ {\mathbb N}}$. Identities (55) and (56) prove the theorem partially with $\alpha = r$ and $\beta = 2r$.

Finally, in (54), by using the replacements $s\to r$ and $2m\to 2m+r$, we have

for $m, r\in{ {\mathbb N}}$, which proves the theorem of the product of $a_{2m+2r}$ and $4a_{2m+r}a_{2m+2r}a_{2m+3r}$ increased by $(a_{\alpha}a_\beta)^2$ with $\alpha = \beta = r$. Thus, the proof of the theorem is completed by the identities (51)-(53) and (55)-(57).

From the Diophantine quadruple (48) of the sequence of Chebyshev polynomials of the first kind we have the following Diophantine quadruple of recursive sequences.

Theorem 4.2. Let $\{ a_n\}_{n\geq 0}$ be a sequence defined by $a_{n} = 2pa_{n-1} +a_{n-2}$, $n\geq 2$, with $a_0 = 1$ and $a_1 = p$. Then, we have a Diophantine quadruple

for $m, r\in{ {\mathbb N}}$. More precisely, the product of any two distinct elements of the set increased by $((a_\alpha-a_\beta)/2)^t$ for suitable integers $\beta>\alpha \geq 0$ is a perfect square.

Proof. The transformation formula (8) shown in Theorem 1.2 with $x_0 = \pm ip/(2\sqrt{q}) = \pm ip/2$ and $2pa_0 = 2p = 2a_1$ gives

Replacing $n$ and $r$ in (2.3)-(2.8) in [26], we have

By substituting $x = x_0 = \pm ip/2$ and then (59) into above identities (60)-(65), we obtain

The proof is completed by (66)-(71).

For $p = 3$, the corresponding sequence $(a_n)_{n\geq 0} = (1, 3, 19,117,721, \ldots)$ is the OEIS sequence A005667 (cf. [27]), which generates a Diophantine quadruple shown in Theorem 4.2.

For many other $p$, we may obtain some new sequences with a Diophantine quadruple. Here, the new sequences mean that they are not included by the OEIS [27].

Acknowledgments

The authors wish to thank Dr. Mark Saul, Dr. and Editor Shouchuan Hu, Dr. and Editor Xingping Sun, and the referees for their carful reading of the draft as well as helpful comments and suggestions.