New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences

Abstract

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities $A_n^2-A_{n+1}{A}_{n-1}=k^{-2n}$, $A_n^2-A_n{A}_{n-1}+{\displaystyle \frac{1}{k^2}}{A}_{n-1}^2=k^{-2n}$, and $B_n^2-B_n{B}_{n-1}+{\displaystyle \frac{1}{k^2}}{B}_{n-1}^2=-(k^2-4)k^{-2n}$, and some generalizations such as $B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2$, $A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$, and more were derived, where $m,n,t\in \mathbb{Z}$ and $t\ne n$. In addition to this, the solution pairs of the algebraic equations $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$, $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$, and $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$ are presented, where $A_p$ and $B_p$ are k-Oresme and k-Oresme–Lucas numbers, respectively.

1. Introduction

Fibonacci and Lucas sequences and their various generalizations are used to model systems such as population dynamics in nature [1], as well as to develop more complex predictive models for stock market analysis and investment strategies [2]. Moreover, these sequences are used to generate cryptographic keys [3], data compression algorithms [4], error correction codes [5], dynamic programming techniques [6], and even signal processing, as well as in the design of non-linear optimization problems [7]. They can also be a source of inspiration for creating rhythmic structures in musical composition [8].

In addition to these applications of Fibonacci and Lucas number sequences, various identities are obtained in this study by performing some matrix operations on k-Oresme and k-Oresme–Lucas number sequences. Then, using these identities, some solutions of Diophantine-like algebraic equations whose coefficients are k-Oresme and k-Oresme–Lucas numbers are provided. First, an overview of the relevant literature is provided in this section.

The basic properties of the k-Fibonacci sequence and its relation to the Lucas numbers were first defined by Falcon and Plaza [9]. There are various studies on the applications of these sequences in combinatorics and numerical analysis [10,11].

Besides these studies, recent studies have extended the algebraic structure and combinatorial properties of the Fibonacci sequence and its various generalizations, such as bronze Fibonacci sequences, hyperbolic quaternions, Fibonacci hybrinomials, and generalized Fibonacci polynomial spinors [12,13,14]. In summary, these studies highlight the versatility and depth of generalized Fibonacci structures in contemporary mathematics research.

Due to their recursive structure, Oresme number sequences attract the interest of researchers in the areas of algebra and analysis of mathematical structures. The Oresme number sequence was first studied in the fourteenth century by Nicole Oresme, a French philosopher, mathematician, physicist, economist, and astronomer. Later, Horadam utilized Oresme’s work to define the Oresme sequence, a sequence with rational coefficients [15].

There are many studies that involve Oresme numbers and their variations. In 2019, Morales provided well-known identities related to Oresme polynomials using matrix methods and the derivatives of these polynomials [16]. In [17], Goy and Zatorsky considered determinants of Toeplitz–Hessenberg matrices with Oresme number entries. Additionally, Aktaş and Soykan found the Frobenius row and column norms of Toeplitz matrices with Oresme number components [18].

In [19], Liana and Wloch defined the Oresme hybrid numbers based on the known Oresme sequence and gave some properties of Oresme hybrid numbers. Based on Liana and Wloch’s work, Halıcı and Sayın analyzed some algebraic properties and applications of these numbers and provided some summation formulas for k-Oresme hybrid numbers [20]. In addition to these, Halıcı and Gür obtained some summation formulas using the k-Oresme polynomial and its derivative [21]. In another study, a geometric approach to Oresme numbers was established by Sayın and Halıcı. Moreover, they defined k-Oresme polynomials with negative indices as a generalization of k-Oresme numbers with negative indices [22,23]. In 2024, Spreafico and Catarino analyzed the connections between generalized Oresme numbers and generalized Fibonacci numbers, providing insight into their recursive properties and matrix representations [24]. Moreover, Demirtürk derived some summation formulas with the help of the first and second derivatives of two polynomials with k-Oresme and k-Oresme–Lucas number coefficients [25].

In the following section, the fundamental definitions and some corollaries of these number sequences are determined. In the third section, matrix operations of k-Fibonacci and k-Oresme sequences are examined. In the fourth section, some new identities related to k-Oresme and k-Oresme–Lucas numbers are provided. Finally, in the last section, solutions of some Diophantine-like equations are determined.

2. Definitions and Main Theorems of k-Fibonacci and k-Oresme Sequences

In this section, definitions and the basic algebraic properties of the k-Oresme and k-Oresme–Lucas sequences are provided, and their relationships to the generalized Fibonacci and generalized Lucas sequences are described. Among the main contributions of this study are the derivation of new sum formulas and combinatorial identities for these special number sequences.

In terms of practical advantages, the results obtained in this study may enable the development of new approaches in areas such as finding the roots of some hyperbolic equations, which have many applications in numerical analysis, cryptography, optimization algorithms, and engineering problems.

Let us provide the definitions of Horadam, k-Fibonacci, and k-Lucas sequences, which belong to the Horadam sequence family.

Definition 1.

Let $a,b,p,q\in \mathbb{Z}$ and $p,q$ be non-zero numbers. The Horadam sequence $W_n=W_n(a,b;p,q)$ is defined by the recurrence relationship $W_n=p{W}_{n-1}+q{W}_{n-2} \mathrm{for} \mathrm{any} n\ge 2$ with the initial conditions $W_0=a, W_1=b$ [26].

Definition 2.

Let $k\ge 3$ be an integer. The k-Fibonacci sequence $\left\{u_n\right\}$ is defined by $u_0=0, u_1=1$, and $u_n=k{u}_{n-1}-u_{n-2}$ for any $n\ge 2$. The k-Lucas sequence $\left\{v_n\right\}$ is defined by $v_0=2, v_1=k$, and $v_n=k{v}_{n-1}-v_{n-2}$ for any $n\ge 2$. $u_n$ and $v_n$ are called the $nth$ k-Fibonacci and Lucas numbers, respectively [27].

Moreover, it can be seen that the $\left\{u_n\right\}$ sequence is $W_n=W_n(0,1;k,-1)$ and the $\left\{v_n\right\}$ sequence is $W_n=W_n(2,k;k,-1)$.

The characteristic equation of these sequences is $x^2-kx+1=0$, with the roots $\alpha ={\displaystyle \frac{k+\sqrt{k^2-4}}{2}}$ and $\beta ={\displaystyle \frac{k-\sqrt{k^2-4}}{2}}$, where $k^2-4>0$. It can easily be seen that $\alpha +\beta =k, \alpha -\beta =\sqrt{k^2-4}, \alpha \beta =1$. Moreover, in [27,28], Binet’s formulas for these sequences are defined as

$$u_n={\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }} \mathrm{and} v_n={\alpha }^n+{\beta }^n, \mathrm{for} \mathrm{all}n\in \mathbb{Z}.$$

As a consequence of Binet’s formulas, these numbers can be extended to negative indices. For all $n\in \mathbb{Z}$, it can be shown that $u_{-n}=-u_n{ \mathrm{and} \mathrm{v}}_{-n}=v_n$ [27]. The properties of these sequences have been studied by many authors [10,11,27,28,29,30].

The Oresme sequence, named after the fourteenth-century mathematician Nicole Oresme, is a mathematical sequence that arises in the study of series, particularly in connection with the harmonic series. Oresme is historically credited with providing one of the earliest demonstrations that the harmonic series diverges. Oresme’s proof of the divergence of the harmonic series is considered a landmark in mathematical history, as it predates many other significant developments in calculus and analysis. Although Oresme’s name is not directly attached to a specific mathematical sequence beyond his work on the harmonic series, his contributions are foundational to the understanding of series, particularly in areas like number theory, summability, and logarithmic growth.

Definition 3.

The Oresme sequence ${\left\{O_n\right\}}_{n\ge 0}$ is defined by the recurrence relationship $O_{n+2}=O_{n+1}-{\displaystyle \frac{1}{4}}{O}_n$ for any $n\ge 0$, with the initial conditions $O_0=0,O_1={\displaystyle \frac{1}{2}}$ [31,32].

Definition 4.

The Oresme–Lucas sequence ${\left\{O{L}_n\right\}}_{n\ge 0}$ is defined by the recurrence relationship $O{L}_{n+2}=O{L}_{n+1}-{\displaystyle \frac{1}{4}}O{L}_n$ for all $n\ge 0$, with the initial conditions $O{L}_0=2,O{L}_1=1$ [31,32].

Özkan and Akkuş defined the k-Oresme and k-Oresme–Lucas sequences in [33] as follows:

Definition 5.

Let $k\ge 3$ be an integer. The k-Oresme sequence $\left\{A_n\right\}$ is provided by the recurrence relationship $A_{n+2}=A_{n+1}-{\displaystyle \frac{1}{k^2}}{A}_n$ for all $n\ge 0$, with the initial conditions $A_0=0,A_1={\displaystyle \frac{1}{k}}.$

Definition 6.

Let $k\ge 3$ be an integer. The k-Oresme–Lucas sequence $\left\{B_n\right\}$ is defined by the recurrence relationship $B_{n+2}=B_{n+1}-{\displaystyle \frac{1}{k^2}}{B}_n$ for all $n\ge 0$, with the initial conditions $B_0=2,B_1=1.$

The characteristic equation of these sequences is $x^2-x+{\displaystyle \frac{1}{k^2}}=0$, so the real roots of this equation are ${\alpha }^{\ast }={\displaystyle \frac{k+\sqrt{k^2-4}}{2k}}$ and ${\beta }^{\ast }={\displaystyle \frac{k-\sqrt{k^2-4}}{2k}}$, where $k^2-4>0.$ It is seen from here that $\alpha =k{\alpha }^{\ast } \mathrm{and} \beta =k{\beta }^{\ast }$.

Binet’s formulas for these sequences are defined as $A_n={\displaystyle \frac{{\left({\alpha }^{\ast }\right)}^n-{\left({\beta }^{\ast }\right)}^n}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k}}$ and $B_n={\left({\alpha }^{\ast }\right)}^n+{\left({\beta }^{\ast }\right)}^n$ for all $n\in \mathbb{N}$ and $k^2-4>0$ [33]. Additionally, Sayin and Halıcı defined the negative indexed k-Oresme and k-Oresme–Lucas numbers [23].

Proposition 1.

For all $n\in \mathbb{N}$, $A_{-n}=-k^n{u}_n=-k^{2n}{A}_n \mathrm{and} B_{-n}=k^n{v}_n=k^{2n}{B}_n$.

Using the relation $\alpha$ between ${\alpha }^{\ast }$ and $\beta$ between ${\beta }^{\ast }$, Proposition 2 can be provided as follows:

Proposition 2.

For all $n\in \mathbb{N}$, $A_n={\left({\displaystyle \frac{1}{k}}\right)}^n{u}_n \mathrm{and} B_n={\left({\displaystyle \frac{1}{k}}\right)}^n{v}_n$.

Thus, we can extend Binet’s formulas for all $n\in \mathbb{Z}$.

Proposition 3.

For all $n\in \mathbb{Z}$, $A_n={\displaystyle \frac{{\left({\alpha }^{\ast }\right)}^n-{\left({\beta }^{\ast }\right)}^n}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k}}$ and $B_n={\left({\alpha }^{\ast }\right)}^n+{\left({\beta }^{\ast }\right)}^n$, with $k^2-4>0$.

Proof of Proposition 3.

Let $n\in \mathbb{N}$. Using the fact that ${\alpha }^{\ast }{\beta }^{\ast }={\displaystyle \frac{1}{k^2}}$, we have

$${\displaystyle \frac{{\left({\alpha }^{\ast }\right)}^{-n}-{\left({\beta }^{\ast }\right)}^{-n}}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k}}={\displaystyle \frac{{\left(\frac{1}{{\alpha }^{\ast }}\right)}^n-{\left(\frac{1}{{\beta }^{\ast }}\right)}^n}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k}}={\displaystyle \frac{{\left({\beta }^{\ast }\right)}^n-{\left({\alpha }^{\ast }\right)}^n}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k{\left({\alpha }^{\ast }{\beta }^{\ast }\right)}^n}}=-k^{2n}\left[{\displaystyle \frac{{\left({\alpha }^{\ast }\right)}^n-{\left({\beta }^{\ast }\right)}^n}{\left({\alpha }^{\ast }-{\beta }^{\ast }\right)k}}\right]=-k^{2n}{A}_n=A_{-n}$$

and

$${\left({\alpha }^{\ast }\right)}^{-n}+{\left({\beta }^{\ast }\right)}^{-n}={\left({\displaystyle \frac{1}{{\alpha }^{\ast }}}\right)}^n+{\left({\displaystyle \frac{1}{{\beta }^{\ast }}}\right)}^n={\displaystyle \frac{{\left({\alpha }^{\ast }\right)}^n+{\left({\beta }^{\ast }\right)}^n}{{\left({\alpha }^{\ast }{\beta }^{\ast }\right)}^n}}=k^{2n}\left[{\left({\alpha }^{\ast }\right)}^n+{\left({\beta }^{\ast }\right)}^n\right]=k^{2n}{B}_n=B_{-n}.$$

Also, taking $n=0$ in Binet’s formulas, we obtain the initial conditions $A_0=0$ and $B_0=2.$

This shows that Binet’s formulas are also satisfied for all negative integers, and $n=0$. Thus, we can use Binet’s formulas for all $n\in \mathbb{Z}$. □

We define the terms with negative indices of k-Oresme and k-Oresme–Lucas sequences in Proposition 4. We omit the proof of Proposition 4, since it follows on from Propositions 2 and 3.

Proposition 4.

For all $n\in \mathbb{Z}$, $A_{-n}=-k^n{u}_n=-k^{2n}{A}_n \mathrm{and} B_{-n}=k^n{v}_n=k^{2n}{B}_n$.

From now on, it is obvious from Propositions 1, 2, and 3 that

$$A_{n+2}=A_{n+1}-{\displaystyle \frac{1}{k^2}}{A}_n\mathrm{and} B_{n+2}=B_{n+1}-{\displaystyle \frac{1}{k^2}}{B}_n\mathrm{for} \mathrm{all} n\in \mathbb{Z}.$$

In the following section, matrices whose elements are k-Oresme and k-Oresme–Lucas numbers are defined, and new identities for these number sequences are obtained using these matrices.

3. Matrix-Based Analysis of k-Fibonacci and k-Oresme Numbers

In this section, several matrix representations and various equations related to k-Fibonacci and k-Oresme numbers are derived. While these topics have been widely studied and discussed in numerous sources, the approach presented here offers a distinct perspective. By employing innovative methods, new relationships and properties are established, contributing a new viewpoint to the existing body of knowledge. This exploration not only enhances the understanding of k-Fibonacci and k-Oresme numbers but also opens new avenues for further research in this area. Now, we can provide some matrix equations in the following theorems.

Theorem 1.

Let $X$ be a square matrix such that $X^2=kX-I$, where $I$ is the identity matrix. Then, for all $n\in \mathbb{Z}$, the following relationship holds:

$$X^n=u_{n}X-u_{n-1}I.$$

Proof of Theorem 1.

For $n=0$, it is trivial as $X^0=I$, which satisfies the relationship $X^0=u_{0}X-u_{-1}I$.

For $n\ge 1$, induction can be used to show that the formula holds.

Let $Y=X^{-1}=kI-X$, which implies that

$$Y^2=k^{2}I-2kX+X^2=k^{2}I-2kX+kX-I=k(kI-X)-I=kY-I.$$

For $n<0$, it follows that

$$X^{-n}=Y^n=u_n(kI-X)-u_{n-1}I=u_{n+1}I-u_{n}X=u_{-n}X-u_{-n-1}I.$$

Thus, the relationship is valid for all $n\in \mathbb{Z}$. □

Theorem 2.

A $2\times 2$ matrix $X$ satisfies the equation $X^2=kX-I$ if and only if $X$ has one of the following forms:

$$X=\left(\begin{array}{cc}a& b\\ c& k-a\end{array}\right) \mathit{with} \det X=1$$

or

$$X=\omega I, \mathrm{where} \omega \in \left\{\alpha ,\beta \right\}.\square$$

We can omit the proof of Theorem 2 since it is provided in [34].

Corollary 1 follows from Theorems 1 and 2.

Corollary 1.

If $X=\left(\begin{array}{cc}a& b\\ c& k-a\end{array}\right)$ is a matrix with $\det X=1$, then $X^n=\left(\begin{array}{cc}a{u}_n-u_{n-1}& b{u}_n\\ c{u}_n& u_{n+1}-a{u}_n\end{array}\right)$ for all $n\in \mathbb{Z}$.

Corollary 2 follows from Theorem 2.

Corollary 2.

${\alpha }^n=\alpha u_n-u_{n-1} \mathrm{and} {\beta }^n=\beta u_n-u_{n-1}$ for all $n\in \mathbb{Z}$.

As a consequence of Corollary 1, Corollary 3 can be provided.

Corollary 3.

Let $Q=\left(\begin{array}{cc}k& 1\\ -1& 0\end{array}\right)$. Then $Q^n=\left(\begin{array}{cc}{u}_{n+1}& u_n\\ -u_n& -u_{n-1}\end{array}\right)$ for all $n\in \mathbb{Z}.$

Corollary 4 follows from Corollary 3.

Corollary 4.

$u_n^2-k{u}_n{u}_{n-1}+u_{n-1}^2=1$ for all $n\in \mathbb{Z}.$

As a result of Corollary 4, Corollary 5 can be provided as follows.

Corollary 5.

Let $A=\left(\begin{array}{cc}{\displaystyle \frac{k}{2}}& {\displaystyle \frac{k^2-4}{2}}\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{k}{2}}\end{array}\right)$. Then $A^n=\left(\begin{array}{cc}{\displaystyle \frac{v_n}{2}}& {\displaystyle \frac{(k^2-4)u_n}{2}}\\ {\displaystyle \frac{u_n}{2}}& {\displaystyle \frac{v_n}{2}}\end{array}\right)$ for all $n\in \mathbb{Z}$.

Taking the determinant of matrix A, we can provide Corollary 6.

Corollary 6.

$v_n^2-\left(k^2-4\right)u_n^2=4$ for all $n\in \mathbb{Z}$.

Lemma 1.

Let $R=\left(\begin{array}{cc}1& {\displaystyle \frac{1}{k}}\\ {\displaystyle \frac{-1}{k}}& 0\end{array}\right)$. Then $R^n=\left(\begin{array}{cc}k{A}_{n+1}& A_n\\ -A_n& {\displaystyle \frac{-1}{k}}{A}_{n-1}\end{array}\right)$ for all $n\in \mathbb{Z}.$

Proof of Lemma 1.

Let us prove this theorem using the induction method. Using the facts that $A_0=0,A_1={\displaystyle \frac{1}{k}}$, and $A_2={\displaystyle \frac{1}{k}}$, it is obvious that the assertion is true for $n=1$. Assume that the equation holds for all $t\in \mathbb{N}$, such that $t<n$, namely, $R^t=\left(\begin{array}{cc}k{A}_{t+1}& A_t\\ -A_t& {\displaystyle \frac{-1}{k}}{A}_{t-1}\end{array}\right)$. If we make the following matrix calculations considering $A_{n+2}=A_{n+1}-{\displaystyle \frac{1}{k^2}}{A}_n$, and

$$\begin{array}{l}R{R}^t=R^{t}R=\left(\begin{array}{cc}k{A}_{t+1}& A_t\\ -A_t& {\displaystyle \frac{-1}{k}}{A}_{t-1}\end{array}\right)\left(\begin{array}{cc}1& {\displaystyle \frac{1}{k}}\\ {\displaystyle \frac{-1}{k}}& 0\end{array}\right)=\left(\begin{array}{cc}k{A}_{t+1}-{\displaystyle \frac{1}{k}}{A}_t& A_{t+1}\\ -A_t+{\displaystyle \frac{1}{k^2}}{A}_{t-1}& {\displaystyle \frac{-1}{k}}{A}_t\end{array}\right)\\ =\left(\begin{array}{cc}k\left(A_{t+1}-{\displaystyle \frac{1}{k^2}}{A}_t\right)& A_{t+1}\\ -\left(A_t-{\displaystyle \frac{1}{k^2}}{A}_{t-1}\right)& {\displaystyle \frac{-1}{k}}{A}_t\end{array}\right)=\left(\begin{array}{cc}k{A}_{t+2}& A_{t+1}\\ -A_{t+1}& {\displaystyle \frac{-1}{k}}{A}_t\end{array}\right)=R^{t+1},\end{array}$$

then we confirm that the equation holds for k + 1. Thus, it follows that the assertion is correct for all $n\in \mathbb{N}$. □

Corollary 7.

For all $n\in \mathbb{Z}$, $A_n^2-A_{n+1}{A}_{n-1}=k^{-2n}$.

Proof of Corollary 7.

The proof is obvious from the determinant of the matrix R. □

Corollary 8 is a consequence of Corollaries 1 and 5.

Corollary 8.

 Let $M=\left(\begin{array}{cc}{\displaystyle \frac{k}{2}}& {\displaystyle \frac{k(k^2-4)}{2}}\\ {\displaystyle \frac{k}{2}}& {\displaystyle \frac{k}{2}}\end{array}\right)$. Then $M^n=k^n\left(\begin{array}{cc}{\displaystyle \frac{B_n}{2}}& {\displaystyle \frac{(k^2-4)A_n}{2}}\\ {\displaystyle \frac{A_n}{2}}& {\displaystyle \frac{B_n}{2}}\end{array}\right)$ for all $n\in \mathbb{N}.$

Theorem 3.

Assume that $a,b, \mathrm{and} a+b$ are nonzero integers. Then

$${\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r}=-{\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r}$$

and

$${\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r-1}={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r-1}$$

for all $n\in \mathbb{Z} \mathrm{and} r\in \mathbb{Z}.$

Proof of Theorem 3.

Let $\mathbb{Z}[\alpha ]=\left\{a\alpha -b\left| a,b\in \mathbb{Z}\right.\right\}.$ Since $\mathbb{Z}[\alpha ]$ is the algebraic integer ring of the real quadratic field $\mathbb{Q}\left(\sqrt{k^2-4}\right)$, the transformation of $\psi :\mathbb{Z}[\alpha ]\to \mathbb{Z}[\alpha ]$ provided by $\psi \left(a\alpha -b\right)=a\beta -b=a\alpha +ak-b$ is a ring isomorphism. Thus, it follows that

$$\psi \left(\alpha u_m-u_{m-1}\right)=\psi \left({\alpha }^m\right)={\left(\psi \left(\alpha \right)\right)}^m={\left(k-\alpha \right)}^m={\beta }^m={\alpha }^{-m}=\alpha u_{-m}-u_{-m-1}=-\alpha u_m+u_{m+1}.$$

$$\begin{array}{ll}\psi \left({\left(a\alpha -b\right)}^n{\alpha }^r\right)& =\psi \left({\left(a\alpha -b\right)}^n\right)\psi \left({\alpha }^r\right)={\left(-a\alpha +(ak-b)\right)}^n{\alpha }^{-r}\\ & ={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{\alpha }^{i-r}\\ & ={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}\left(\alpha u_{i-r}-u_{i-r-1}\right)\\ & =\alpha \left({\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r}\right)-\left({\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r-1}\right)\end{array}$$

and

$$\begin{array}{ll}\psi \left({\left(a\alpha -b\right)}^n{\alpha }^r\right)& =\psi \left(\left({\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{\alpha }^i\right){\alpha }^r\right)=\psi \left({\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{\alpha }^{i+r}\right)\\ & =\psi \left({\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}\left(\alpha u_{i+r}-u_{i+r-1}\right)\right)\\ & =\alpha \left(-{\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r}\right)-{\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r-1}.\end{array}$$

Therefore, we have

$${\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r}=-{\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r}$$

and

$${\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{\left(-a\right)}^i{(ak-b)}^{n-i}{u}_{i-r-1}={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{a}^i{(-b)}^{n-i}{u}_{i+r-1}.$$

□

According to Theorem 3, Corollary 9 can be provided.

Corollary 9.

Let $m,r\in \mathbb{Z}$, with $m\ne 1 \mathrm{and} m\ne 0.$ For all $n\in \mathbb{N}$, we have

$$u_{mn+r}={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{u}_m^i{(-u_{m-1})}^{n-i}{u}_{i+r},$$

$$v_{mn+r}={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{u}_m^i{(-u_{m-1})}^{n-i}{v}_{i+r},$$

$$u_{mn+r}=-{\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{(-u_m)}^i{u}_{m+1}^{n-i}{u}_{i-r},$$

and

$$v_{mn+r}={\displaystyle \sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)}{(-u_m)}^i{u}_{m+1}^{n-i}{v}_{i-r}.$$

Theorem 4.

 Let $m,r\in \mathbb{Z}$ and $n\in \mathbb{N}$, then we have

$${(k^2-4)}^n{u}_{2nm+r}={\displaystyle \sum _{i=0}^{2n}\left(\begin{array}{c}2n\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n-i}{u}_{i+r},$$

$${(k^2-4)}^n{v}_{2nm+r}={\displaystyle \sum _{i=0}^{2n}\left(\begin{array}{c}2n\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n-i}{v}_{i+r},$$

$${(k^2-4)}^{n+1}{u}_{(2n+1)m+r}={\displaystyle \sum _{i=0}^{2n+1}\left(\begin{array}{c}2n+1\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n+1-i}{v}_{i+r},$$

and

$${(k^2-4)}^n{v}_{(2n+1)m+r}={\displaystyle \sum _{i=0}^{2n+1}\left(\begin{array}{c}2n+1\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n+1-i}{u}_{i+r}.$$

Proof of Theorem 4.

Utilizing the relationships

$$\left(\begin{array}{cc}k& 2\\ -2& -k\end{array}\right)\left(\begin{array}{cc}{u}_{m+1}& u_m\\ -u_m& -u_{m-1}\end{array}\right)=\left(\begin{array}{cc}{u}_{m+1}& u_m\\ -u_m& -u_{m-1}\end{array}\right)\left(\begin{array}{cc}k& 2\\ -2& -k\end{array}\right)=\left(\begin{array}{cc}{v}_{m+1}& v_m\\ -v_m& -v_{m-1}\end{array}\right)$$

and

$${\left(\begin{array}{cc}k& 2\\ -2& -k\end{array}\right)}^2=(k^2-4)I,$$

we arrive at

$$\begin{array}{c}{(k^2-4)}^n{Q}^{2mn+r}={(k^2-4)}^n\left(\begin{array}{cc}{u}_{2mn+1}& u_{2mn}\\ -u_{2mn}& -u_{2mn-1}\end{array}\right)Q^r={\left(\begin{array}{cc}{v}_{m+1}& v_m\\ -v_m& -v_{m-1}\end{array}\right)}^{2n}{Q}^r\\ ={\left(v_{m}Q-v_{m-1}I\right)}^{2n}{Q}^r={\displaystyle \sum _{i=0}^{2n}\left(\begin{array}{c}2n\\ i\end{array}\right)v_m^i}{(-v_{m-1})}^{2n-i}{Q}^{i+r}.\end{array}$$

Thus, it follows that

$${(k^2-4)}^n{u}_{2nm+r}={\displaystyle \sum _{i=0}^{2n}\left(\begin{array}{c}2n\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n-i}{u}_{i+r},$$

$${(k^2-4)}^n{v}_{2nm+r}={\displaystyle \sum _{i=0}^{2n}\left(\begin{array}{c}2n\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n-i}{v}_{i+r},$$

$${(k^2-4)}^{n+1}{u}_{(2n+1)m+r}={\displaystyle \sum _{i=0}^{2n+1}\left(\begin{array}{c}2n+1\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n+1-i}{v}_{i+r},$$

and

$${(k^2-4)}^n{v}_{(2n+1)m+r}={\displaystyle \sum _{i=0}^{2n+1}\left(\begin{array}{c}2n+1\\ i\end{array}\right)}{v}_m^i{(-v_{m-1})}^{2n+1-i}{u}_{i+r}.$$

□

Moreover, we can provide some identities concerning k-Oresme and k-Oresme–Lucas sequences based on Corollaries 3, 4, 5, 6, and 8 and Lemma 1. These identities will be useful in the next section when obtaining new equations related to k-Oresme and k-Oresme–Lucas numbers. Since these identities can be shown using Binet’s formulas, we have omitted their proofs.

$$A_n^2-A_n{A}_{n-1}+{\displaystyle \frac{1}{k^2}}{A}_{n-1}^2=k^{-2n}$$

(1)

$$B_n^2-B_n{B}_{n-1}+{\displaystyle \frac{1}{k^2}}{B}_{n-1}^2=-(k^2-4)k^{-2n}$$

(2)

$$B_n{A}_m-A_n{B}_m=2k^{-2n}{A}_{m-n}$$

(3)

$$B_n{B}_m-(k^2-4)A_n{A}_m=2k^{-2n}{B}_{m-n}$$

(4)

$$B_n{B}_m+(k^2-4)A_n{A}_m=2B_{n+m}$$

(5)

$$A_n{A}_{m+1}-{\displaystyle \frac{1}{k^2}}{A}_m{A}_{n-1}={\displaystyle \frac{1}{k}}{A}_{n+m}$$

(6)

$$A_n{B}_m+B_n{A}_m=2A_{n+m}$$

(7)

$$k{A}_{n+1}-{\displaystyle \frac{1}{k}}{A}_{n-1}=B_n$$

(8)

$$B_{n+1}-{\displaystyle \frac{1}{k^2}}{B}_{n-1}={\displaystyle \frac{k^2-4}{k}}{A}_n$$

(9)

$$B_n^2-(k^2-4)A_n^2=4k^{-2n}$$

(10)

$$A_{n+1}{B}_m-{\displaystyle \frac{1}{k^2}}{A}_n{B}_{m-1}={\displaystyle \frac{1}{k}}{B}_{n+m}$$

(11)

4. New Identities Through Matrix Characterization

We can provide some new equations using the matrix $M^n$ provided in Corollary 8 and the identities provided in (1)–(11).

Theorem 5.

For all $m,n,t\in \mathbb{Z}$,

$$B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2.$$

(12)

Proof of Theorem 5.

The matrix multiplication and Equations (5) and (7) provide

$$\left(\begin{array}{cc}{\displaystyle \frac{B_n}{2}}& {\displaystyle \frac{(k^2-4)A_n}{2}}\\ {\displaystyle \frac{A_t}{2}}& {\displaystyle \frac{B_t}{2}}\end{array}\right)\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{B_n{B}_m+(k^2-4)A_n{A}_m}{2}}\\ {\displaystyle \frac{A_t{B}_m+B_t{A}_m}{2}}\end{array}\right)=\left(\begin{array}{c}{B}_{n+m}\\ A_{t+m}\end{array}\right).$$

Also, using the determinant of the following matrix with Equation (4), we get

$$\left|\begin{array}{cc}{\displaystyle \frac{B_n}{2}}& {\displaystyle \frac{(k^2-4)A_n}{2}}\\ {\displaystyle \frac{A_t}{2}}& {\displaystyle \frac{B_t}{2}}\end{array}\right|={\displaystyle \frac{B_n{B}_t-(k^2-4)A_n{A}_t}{4}}={\displaystyle \frac{k^{-2t}{B}_{n-t}}{2}}\ne 0.$$

It follows that

$$\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)={\displaystyle \frac{2k^{2t}}{B_{t-n}}}\left(\begin{array}{cc}{\displaystyle \frac{B_t}{2}}& {\displaystyle \frac{-(k^2-4)A_n}{2}}\\ {\displaystyle \frac{-A_t}{2}}& {\displaystyle \frac{B_n}{2}}\end{array}\right)\left(\begin{array}{c}{B}_{n+m}\\ A_{t+m}\end{array}\right).$$

Hence, we get

$$\begin{array}{l}{B}_m={\displaystyle \frac{2k^{2t}\left(B_t{B}_{n+m}-(k^2-4)A_n{A}_{t+m}\right)}{2B_{n-t}}}={\displaystyle \frac{k^{2t}\left(B_t{B}_{n+m}-(k^2-4)A_n{A}_{t+m}\right)}{B_{n-t}}}\\ A_m={\displaystyle \frac{2k^{2t}\left(B_n{A}_{t+m}-A_t{B}_{n+m}\right)}{2B_{n-t}}}={\displaystyle \frac{k^{2t}\left(B_n{A}_{t+m}-A_t{B}_{n+m}\right)}{B_{n-t}}}.\end{array}$$

Using Equations (10) and (3), it follows that

$$\begin{array}{l}4k^{-2m}=B_m^2-(k^2-4)A_m^2={\displaystyle \frac{{\left(k^{2t}\right)}^2\left[{\left(B_t{B}_{n+m}-(k^2-4)A_n{A}_{t+m}\right)}^2-(k^2-4){\left(B_n{A}_{t+m}-A_t{B}_{n+m}\right)}^2\right]}{B_{n-t}^2}}\\ ={\displaystyle \frac{k^{4t}\left[\left(B_t^2{B}_{n+m}^2-2(k^2-4)B_t{B}_{n+m}{A}_n{A}_{t+m}+{(k^2-4)}^2{A}_n^2{A}_{t+m}^2\right)-(k^2-4)\left(B_n^2{A}_{t+m}^2-2B_n{A}_{t+m}{A}_t{B}_{n+m}+A_t^2{B}_{n+m}^2\right)\right]}{B_{n-t}^2}}\\ ={\displaystyle \frac{k^{4t}\left[B_{n+m}^2\left(B_t^2-(k^2-4)A_t^2\right)-2(k^2-4)\left(B_t{A}_n-A_t{B}_n\right)B_{n+m}{A}_{t+m}-(k^2-4)\left(B_n^2-(k^2-4)A_n^2\right)A_{t+m}^2\right]}{B_{n-t}^2}}\\ ={\displaystyle \frac{k^{4t}\left[B_{n+m}^{2}4k^{-2t}-2(k^2-4)2k^{-2t}{A}_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)4k^{-2n}{A}_{t+m}^2\right]}{B_{n-t}^2}}\\ ={\displaystyle \frac{4k^{2t}\left[B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2\right]}{B_{n-t}^2}}\end{array}$$

Thus, we get

$$B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2={\displaystyle \frac{4k^{-2m}{B}_{n-t}^2}{4k^{2t}}}=k^{-2m-2t}{B}_{n-t}^2,$$

which provides the expected equation

$$B_{n+m}^2-(k^2-4)A_{n-t}{B}_{n+m}{A}_{t+m}-(k^2-4)k^{2t-2n}{A}_{t+m}^2=k^{-2m-2t}{B}_{n-t}^2.$$

□

Theorem 6.

 For all $m,n,t\in \mathbb{Z}$ and $t\ne n$, we have

$$B_{t+m}^2-B_{t-n}{B}_{n+m}{B}_{t+m}+k^{2n-2t}{B}_{n+m}^2=-(k^2-4)k^{-2m-2n}{A}_{t-n}^2$$

(13)

Proof of Theorem 6.

The matrix multiplication and Equation (5) provide

$$\left(\begin{array}{cc}{\displaystyle \frac{B_n}{2}}& {\displaystyle \frac{(k^2-4)A_n}{2}}\\ {\displaystyle \frac{B_t}{2}}& {\displaystyle \frac{(k^2-4)A_t}{2}}\end{array}\right)\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{B_n{B}_m+(k^2-4)A_n{A}_m}{2}}\\ {\displaystyle \frac{B_t{B}_m+(k^2-4)A_t{A}_m}{2}}\end{array}\right)=\left(\begin{array}{c}2B_{n+m}\\ 2B_{t+m}\end{array}\right).$$

Also, using the determinant with Equation (3), we get

$$\left|\begin{array}{cc}{\displaystyle \frac{B_n}{2}}& {\displaystyle \frac{(k^2-4)A_n}{2}}\\ {\displaystyle \frac{B_t}{2}}& {\displaystyle \frac{(k^2-4)A_t}{2}}\end{array}\right|={\displaystyle \frac{(k^2-4)\left(B_n{A}_t-A_n{B}_t\right)}{4}}={\displaystyle \frac{2k^{-2n}(k^2-4)A_{t-n}}{4}}={\displaystyle \frac{k^{-2n}(k^2-4)A_{t-n}}{2}}\ne 0$$

since $t\ne n$. Thus, it follows that

$$\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)={\displaystyle \frac{2k^{2n}}{(k^2-4)A_{t-n}}}\left(\begin{array}{cc}{\displaystyle \frac{(k^2-4)A_t}{2}}& {\displaystyle \frac{-(k^2-4)A_n}{2}}\\ {\displaystyle \frac{-B_t}{2}}& {\displaystyle \frac{B_n}{2}}\end{array}\right)\left(\begin{array}{c}2B_{n+m}\\ 2B_{t+m}\end{array}\right).$$

Hence, we get

$$\begin{array}{l}{B}_m={\displaystyle \frac{2k^{2n}\left(A_t{B}_{n+m}-A_n{B}_{t+m}\right)}{A_{t-n}}}\\ A_m={\displaystyle \frac{2k^{2n}\left(B_n{B}_{t+m}-B_t{B}_{n+m}\right)}{(k^2-4)A_{t-n}}}.\end{array}$$

Using Equations (4) and (10), it follows that

$$\begin{array}{l}4k^{-2m}=B_m^2-(k^2-4)A_m^2={\displaystyle \frac{4k^{4n}}{A_{t-n}^2}}\left[{\left(A_t{B}_{n+m}-A_n{B}_{t+m}\right)}^2-(k^2-4){\displaystyle \frac{{\left(B_n{B}_{t+m}-B_t{B}_{n+m}\right)}^2}{{(k^2-4)}^2}}\right]\\ ={\displaystyle \frac{4k^{4n}\left[(k^2-4)\left(A_t^2{B}_{n+m}^2-2A_t{B}_{n+m}{A}_n{B}_{t+m}+A_n^2{B}_{t+m}^2\right)-\left(B_n^2{B}_{t+m}^2-2B_n{B}_{t+m}{B}_t{B}_{n+m}+B_t^2{B}_{n+m}^2\right)\right]}{(k^2-4)A_{t-n}^2}}\\ ={\displaystyle \frac{4k^{4n}\left[-B_{n+m}^2\left(B_t^2-(k^2-4)A_t^2\right)+2\left(B_n{B}_t-(k^2-4)A_n{A}_t\right)B_{n+m}{B}_{t+m}-\left(B_n^2-(k^2-4)A_n^2\right)B_{t+m}^2\right]}{(k^2-4)A_{t-n}^2}}\\ ={\displaystyle \frac{4k^{4n}\left[-4k^{-2t}{B}_{n+m}^2+4k^{-2n}{B}_{t-n}{B}_{n+m}{B}_{t+m}-4k^{-2n}{B}_{t+m}^2\right]}{(k^2-4)A_{t-n}^2}}\\ ={\displaystyle \frac{4k^{2n}\left[-k^{-2t+2n}{B}_{n+m}^2+B_{t-n}{B}_{n+m}{B}_{t+m}-B_{t+m}^2\right]}{(k^2-4)A_{t-n}^2}}.\end{array}$$

Therefore, we have

$$\begin{array}{l}4(k^2-4)k^{-2m}{A}_{t-n}^2=4k^{2n}\left[-k^{2n-2t}{B}_{n+m}^2+B_{t-n}{B}_{n+m}{B}_{t+m}-B_{t+m}^2\right]\\ k^{2n-2t}{B}_{n+m}^2-B_{t-n}{B}_{n+m}{B}_{t+m}+B_{t+m}^2={\displaystyle \frac{-4(k^2-4)k^{-2m}{A}_{t-n}^2}{4k^{2n}}}=-(k^2-4)k^{-2m-2n}{A}_{t-n}^2\end{array}$$

which provides the expected equation

$$B_{t+m}^2-B_{t-n}{B}_{n+m}{B}_{t+m}+k^{2n-2t}{B}_{n+m}^2=-(k^2-4)k^{-2m-2n}{A}_{t-n}^2$$

□

Theorem 7.

For all $m,n,t\in \mathbb{Z}$ and $t\ne n$, we have

$$A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$$

(14)

Proof of Theorem 7.

The matrix multiplication and Equation (7) provide

$$\left(\begin{array}{cc}{\displaystyle \frac{A_n}{2}}& {\displaystyle \frac{B_n}{2}}\\ {\displaystyle \frac{A_t}{2}}& {\displaystyle \frac{B_t}{2}}\end{array}\right)\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{A_n{B}_m+B_n{A}_m}{2}}\\ {\displaystyle \frac{A_t{B}_m+B_t{A}_m}{2}}\end{array}\right)=\left(\begin{array}{c}{A}_{n+m}\\ A_{t+m}\end{array}\right).$$

Also, calculating the following determinant with Equation (3), we get

$$\left|\begin{array}{cc}{\displaystyle \frac{A_n}{2}}& {\displaystyle \frac{B_n}{2}}\\ {\displaystyle \frac{A_t}{2}}& {\displaystyle \frac{B_t}{2}}\end{array}\right|={\displaystyle \frac{A_n{B}_t-B_n{A}_t}{4}}={\displaystyle \frac{-k^{-2n}{A}_{t-n}}{2}}\ne 0,$$

since $t\ne n$. It follows that

$$\left(\begin{array}{c}{B}_m\\ A_m\end{array}\right)={\displaystyle \frac{-2k^{2n}}{A_{t-n}}}\left(\begin{array}{cc}{\displaystyle \frac{B_t}{2}}& {\displaystyle \frac{-B_n}{2}}\\ {\displaystyle \frac{-A_t}{2}}& {\displaystyle \frac{A_n}{2}}\end{array}\right)\left(\begin{array}{c}{A}_{n+m}\\ A_{t+m}\end{array}\right).$$

Hence, we get

$$\begin{array}{l}{A}_m={\displaystyle \frac{-k^{2n}\left(A_n{A}_{t+m}-A_t{A}_{n+m}\right)}{A_{t-n}}}\\ B_m={\displaystyle \frac{-k^{2n}\left(B_t{A}_{n+m}-B_n{A}_{t+m}\right)}{A_{t-n}}}\end{array}.$$

Using Equations (4) and (10), it follows that

$$\begin{array}{l}4k^{-2m}=B_m^2-(k^2-4)A_m^2={\displaystyle \frac{k^{4n}}{A_{t-n}^2}}\left[{\left(B_t{A}_{n+m}-B_n{A}_{t+m}\right)}^2-(k^2-4){\left(A_n{A}_{t+m}-A_t{A}_{n+m}\right)}^2\right]\\ ={\displaystyle \frac{k^{4n}\left[4k^{-2t}{A}_{n+m}^2-4k^{-2n}{B}_{t-n}{A}_{n+m}{A}_{t+m}+4k^{-2n}{A}_{t+m}^2\right]}{A_{t-n}^2}}\\ ={\displaystyle \frac{4k^{2n}\left[k^{2n-2t}{A}_{n+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+A_{t+m}^2\right]}{A_{t-n}^2}}.\end{array}$$

Thus, it is seen that

$$A_{t+m}^2-B_{t-n}{A}_{n+m}{A}_{t+m}+k^{2n-2t}{A}_{n+m}^2=k^{-2n-2m}{A}_{t-n}^2$$

□

5. Solutions of Some Equations with k-Oresme and k-Oresme–Lucas Number Coefficients

Diophantine equations are algebraic equations whose variables and coefficients are integers and which seek integer solutions. One of the most well-known examples, Fermat’s Last Theorem, has been a challenging problem that mathematicians have been trying to solve for centuries. Today, Diophantine equations are an active topic of research, attracting many mathematicians in algebra and number theory. For example, some studies in which all integer solutions of Diophantine equations, whose coefficients are Fibonacci and Lucas numbers, are expressed in terms of Fibonacci and Lucas numbers as provided in [35,36,37]. Generalized versions of these Diophantine equations are considered in [38,39].

In this section, different from the above Diophantine equations, the problem of finding solutions of algebraic equations whose coefficients and solutions are not integers is posed by the following three theorems. For the proofs of Theorems 8, 9, and 10, one can consult Equations (12), (13) and (14), respectively.

Theorem 8.

Let $k\ge 3$. Then, the pair $(x,y)=\mp (B_{p+q},A_q)$ satisfies the equation $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$ for all $p,q\in \mathbb{Z}$.

Proof of Theorem 8.

If we consider Theorem 5 and take $n+m=p+q, t+m=q \mathrm{and} n-t=p$ in (12), then we obtain the equation

$$B_{p+q}^2-(k^2-4)A_p{B}_{p+q}{A}_q-(k^2-4)k^{-2p}{A}_q^2=k^{-2q}{B}_p^2,$$

which implies that the pair $(x,y)=\mp (B_{p+q},A_q)$ satisfies the equation $x^2-(k^2-4)A_{p}xy-(k^2-4)k^{-2p}{y}^2=k^{-2q}{B}_p^2$. □

Theorem 9.

Let $k\ge 3$. Then, the pair $(x,y)=\mp (B_{p+q},B_q)$ satisfies the equation $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$ for all $p,q\in \mathbb{Z}$.

Proof of Theorem 9.

Taking $t+m=p+q, t-n=p \mathrm{and} n+m=q$ in (13), then we obtain the equation

$$B_{p+q}^2-B_p{B}_{p+q}{B}_q+k^{-2p}{B}_q^2=-(k^2-4)k^{-2q}{A}_p^2$$

which implies that the pair $(x,y)=\mp (B_{p+q},B_q)$ satisfies the equation $x^2-B_{p}xy+k^{-2p}{y}^2=-(k^2-4)k^{-2q}{A}_p^2$. □

Theorem 10.

The pair $(x,y)=\mp (A_{p+q},A_q)$ satisfies the equation $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$ for all $p,q\in \mathbb{Z}$.

Proof of Theorem 10.

Taking $t+m=p+q, t-n=p \mathrm{and} n+m=q$ in (14), then we obtain the equation

$$A_{p+q}^2-B_p{A}_p{A}_q+k^{-2p}{A}_q^2=k^{-2q}{A}_p^2$$

which implies that the pair $(x,y)=\mp (A_{p+q},A_q)$ satisfies the equation $x^2-B_{p}xy+k^{-2p}{y}^2=k^{-2q}{A}_p^2$. □

6. Conclusions

In this study, matrices whose elements are k-Oresme and k-Oresme–Lucas numbers were considered, and identities related to k-Oresme and k-Oresme–Lucas numbers were obtained. Using these identities, it is claimed in the last three theorems that some solutions of algebraic equations whose coefficients are k-Oresme and k-Oresme–Lucas numbers can be expressed in terms of k-Oresme and k-Oresme–Lucas numbers. In future works, by considering different matrices with number sequence elements, all solutions of different Diophantine-like algebraic equations can be obtained. Moreover, by studying the algebraic properties of these sequences in this way, the obtained results provide potential contributions not only from a theoretical point of view but also in applied mathematics and computer science. In particular, solutions of equations based on number sequences are fundamental for topics such as prime number characterizations, divisibility properties, and the analysis of algebraic structures in number theory. In this respect, this study demonstrates the capability of Oresme-type number sequences in generating algebraic solutions, while showing that these sequences can form the basis for new mathematical approaches.

Due to the contributions of this study, which focuses on the algebraic properties of number sequences, algebraic structures can be better understood, new algorithms using these number sequences can be developed, and performance comparisons can be made. Our future work will be in this direction.