On applications of Pell and Pell-Lucas numbers with matrix method

Abstract

In this study, new matrices which produce the Pell and Pell-Lucas numbers are given. By using these matrices, new identities and relations related to the Pell and Pell-Lucas numbers are obtained.

Keywords

Pell numbers Pell-Lucas numbers matrices

1 Introduction

Integer sequences are one of the most important topics in number theory. In particular, Fibonacci and Pell number sequences are the most interesting and researched. These integer sequences have been used in many applied mathematics fields. It is well known that matrix relations of the sequences have been used in many areas such as coding theory, statistic theory, and computing theory [11 , 15]. The authors in [6 , 9] examined the Fibonacci and Lucas numbers by matrices. The authors contributed to the literature with new properties of these numbers. In [2, 3], Horadam investigated Pell and Pell-Lucas sequences. These number sequences are in a relationship with some recursive formulations and with the n^th power of some 2- dimensional square matrices, these numbers can be obtained like Fibonacci numbers. In [8], Ercolano found the matrices which produce Pell numbers. In [1], Dasdemir compared Pell and Pell-Lucas numbers by using those matrices. In [5], the authors studied the properties of the Pell matrix with linear algebra. In [4], Değer found the relation between Fibonacci numbers and the suborbital graphs with a new matrix that gives even indexed Fibonacci numbers by using that relation. In [13], the authors examined the relationship between suborbital graphs and Pell numbers. Then, they used a new matrix that produces even indexed Pell and Pell-Lucas numbers.

In this study, firstly we give new matrices related to Pell and Pell-Lucas numbers by using matrix properties. We produce new relations with these matrices. Then using the lambda function from [14] and its properties, we give new identities consisting of Pell and Pell-Lucas numbers.

Pell and Pell-Lucas numbers

P_n is the n^th Pell number which satisfies the recurrence relation P_n = 2P_n-1 + P_n-2, n ≥ 2 by initial conditions P₀ = 0 and P₁ = 1. Similarly, Q_n is the n^th Pell-Lucas number by Q_n = 2Q_n-1 + Q_n-2, n ≥ 2 and initial conditions Q₀ = 1, Q₁ = 1. Binet-like formulas for P_n and Q_n are $\frac{γ^{n} - δ^{n}}{γ - δ}$ and $\frac{γ^{n} + δ^{n}}{2}$ , respectively, where $γ = 1 + \sqrt{2}$ , $δ = 1 - \sqrt{2}$ , and n ≥ 2.

For Pell and Pell-Lucas numbers, the following identities are given: $P_{n} + P_{n - 1} = Q_{n},$ (1) $P_{n + 1} + P_{n - 1} = 2 Q_{n},$ (2) $P_{2 n} = 2 P_{n} Q_{n},$ (3) $P_{n} + Q_{n} = P_{n + 1},$ (4) $P_{n + 1} P_{n - 1} - {P_{n}}^{2} = (- 1)^{n},$ (5) $Q_{n + 1} Q_{n - 1} - {Q_{n}}^{2} = 2 (- 1)^{n - 1},$ (6) where $n \in ℤ^{+} .$ See [10] for details.

2 Some new matrices representations of Pell and Pell-Lucas numbers

In this section, we give some new matrices which produce Pell and Pell-Lucas numbers.

Theorem 1. Let P_n is the n^th Pell number and Q_n is the n^th Pell- Lucas number, then we get the following equation as

$Z^{n} = (\begin{matrix} {(- 1)}^{n} P_{n - 1} & 0 & {(- 1)}^{n + 1} P_{n} \\ \frac{k}{2} [{(- 1)}^{n + 1} Q_{n - 1} - 1] & 1 & \frac{k}{2} [{(- 1)}^{n} Q_{n} - 1] \\ {(- 1)}^{n + 1} P_{n} & 0 & {(- 1)}^{n} P_{n + 1} \end{matrix}),$ where $Z = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & - k \\ 1 & 0 & - 2 \end{matrix})$ and $n \in ℤ^{+}$ .

Proof. For the proof we use the diagonal matrix of matrix Z. Because we want to write the n^th power of matrix Zⁿ with the equation,

Dⁿ = (P^-1ZP) ⁿ = P^-1ZPP^-1 ⋯ P^-1ZP = P^-1ZⁿP,

where D is the diagonal matrix, P is the matrix whose column i is i^th eigenvector. Hence, we get Zⁿ = PDⁿP^-1. Now let us find the eigenvalues and eigenvectors of matrix Z. Then by the equation $P_{Z} (λ) = \det (λ I - Z) = λ^{3} - 2 λ + 1,$ we can find the eigenvalues as $λ_{1, 2, 3} = 1, - \sqrt{2} - 1, - 1 + \sqrt{2}$ and the eigenvectors as

$V_{1, 2, 3} = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), (\begin{matrix} 1 - \sqrt{2} \\ k (- \frac{\sqrt{2}}{2} + 1) \\ 1 \end{matrix}), (\begin{matrix} 1 + \sqrt{2} \\ k (\frac{\sqrt{2}}{2} + 1) \\ 1 \end{matrix}) .$

When we write matrix P and its inverse P^-1. Then, $P = (\begin{matrix} 0 & 1 - \sqrt{2} & 1 + \sqrt{2} \\ 1 & k (- \frac{\sqrt{2}}{2} + 1) & k (\frac{\sqrt{2}}{2} + 1) \\ 0 & 1 & 1 \end{matrix})$ and $P^{- 1} = (\begin{matrix} - \frac{k}{2} & 1 & - \frac{k}{2} \\ - \frac{\sqrt{2}}{4} & 0 & \frac{\sqrt{2} + 2}{4} \\ \frac{\sqrt{2}}{4} & 0 & \frac{- \sqrt{2} + 2}{4} \end{matrix}) .$ Therefore, we can write the diagonal matrix as $D = (\begin{matrix} 1 & 0 & 0 \\ 0 & - \sqrt{2} - 1 & 0 \\ 0 & 0 & - 1 + \sqrt{2} \end{matrix}) .$ Now, we can write the matrix Zⁿ as:

Zⁿ = PDⁿP^-1 =

$(\begin{matrix} \frac{(\sqrt{2} + 2) (\sqrt{2} - 1)^{n} - (\sqrt{2} - 2) (- \sqrt{2} - 1)^{n}}{4} & 0 & \frac{\sqrt{2} (\sqrt{2} - 1)^{n} - \sqrt{2} (- \sqrt{2} - 1)^{n}}{4} \\ \frac{(\sqrt{2} + 1) k {(\sqrt{2} - 1)}^{n} - (\sqrt{2} - 1) k {(- \sqrt{2} - 1)}^{n}}{4} - \frac{k}{2} & 1 & \frac{k {(\sqrt{2} - 1)}^{n} - k {(- \sqrt{2} - 1)}^{n}}{4} - \frac{k}{2} \\ \frac{\sqrt{2} (\sqrt{2} - 1)^{n} - \sqrt{2} (- \sqrt{2} - 1)^{n}}{4} & 0 & \frac{(- \sqrt{2} + 2) {(\sqrt{2} - 1)}^{n} + (- \sqrt{2} + 2) {(\sqrt{2} - 1)}^{n}}{4} \end{matrix}) .$

If we use Binet like formula of Pell and Pell-Lucas numbers as $P_{n} = \frac{γ^{n} - δ^{n}}{γ - δ} and Q_{n} = \frac{γ^{n} + δ^{n}}{2}$ , where $γ = 1 + \sqrt{2}$ and $δ = 1 - \sqrt{2}$ , we get the matrix Zⁿ with Pell and Pell-Lucas numbers.

Theorem 2. Let P_n and Q_n are n^th Pell and Pell-Lucas numbers, respectively. Then, for $n \in ℤ^{+}$ ,

$V_{L}^{n} = {\begin{matrix} 2^{\frac{3}{2} n - 1} [\begin{matrix} - P_{2 n - 2} & P_{2 n} \\ - P_{2 n} & P_{2 n + 2} \end{matrix}] foreven n, \\ 2^{\frac{3}{2} (n - 1)} [\begin{matrix} Q_{2 n - 2} & - Q_{2 n} \\ Q_{2 n} & - Q_{2 n + 2} \end{matrix}] forodd n, \end{matrix}$ (7) where $V_{L} = [\begin{matrix} 1 & - 3 \\ 3 & - 17 \end{matrix}] .$

Proof. For the proof, we can use the mathematical induction method on n. Firstly, we take for odd integers n, the case for n = 1 is true. We suppose that for n = k, the case is true and we should show the case is true for odd integer n = k + 2. So,

$\begin{matrix} V_{L}^{k + 2} & = & V_{L}^{k} V_{L}^{2} \\ = & {[\begin{matrix} 1 & - 3 \\ 3 & - 17 \end{matrix}]}^{k} {[\begin{matrix} 1 & - 3 \\ 3 & - 17 \end{matrix}]}^{2} \\ = & 2^{\frac{3}{2} (k - 1)} [\begin{matrix} Q_{2 k - 2} & - Q_{2 k} \\ Q_{2 k} & - Q_{2 k + 2} \end{matrix}] [\begin{matrix} - 8 & 48 \\ - 48 & 280 ∥ \end{matrix}] \\ = & 2^{\frac{3}{2} (k + 1)} [\begin{matrix} Q_{2 k + 2} & - Q_{2 k + 4} \\ Q_{2 k + 4} & - Q_{2 k + 6} ∥ \end{matrix}], \end{matrix}$ where we use recurrence relation for Pell-Lucas numbers. So, the allegation for all odd integers n is valid. For even integers n, the proof can be made similarly.

3 Matrix application of Pell, Pell-Lucas numbers

In this section by using the matrices Zⁿ and $V_{L}^{n}$ , we obtain new identities and relations of Pell and Pell-Lucas numbers.

Theorem 3. The following identities are valid for $m, n \in ℤ^{+}$ ;

P_m+n = P_n-1P_m + P_m+1P_n,

P_m+n = P_nP_m-1 + P_n+1P_m,

Q_m+n = Q_n-1P_m + Q_nP_m+1.

Proof. By using the matrix Zⁿ, we write Z^m+n as;

Z^m+n =

$(\begin{matrix} {(- 1)}^{m + n} P_{m + n - 1} & 0 & {(- 1)}^{m + n + 1} P_{m + n} \\ \frac{k}{2} [{(- 1)}^{n + m + 1} Q_{m + n - 1} - 1] & 1 & \frac{k}{2} [{(- 1)}^{m + n} Q_{m + n} - 1] \\ {(- 1)}^{m + n + 1} P_{m + n} & 0 & {(- 1)}^{m + n} P_{m + n + 1} \end{matrix}) .$

From the matrix properties, if Z^mZⁿ = Z^m+n, then,

Z^mZⁿ =

$(\begin{matrix} {(- 1)}^{m} P_{m - 1} & 0 & {(- 1)}^{m + 1} P_{m} \\ \frac{k}{2} [{(- 1)}^{m + 1} Q_{m - 1} - 1] & 1 & \frac{k}{2} [{(- 1)}^{m} Q_{m} - 1] \\ {(- 1)}^{m + 1} P_{m} & 0 & {(- 1)}^{m} P_{m + 1} \end{matrix}) \times$

$(\begin{matrix} {(- 1)}^{n} P_{n - 1} & 0 & {(- 1)}^{n + 1} P_{n} \\ \frac{k}{2} [{(- 1)}^{n + 1} Q_{n - 1} - 1] & 1 & \frac{k}{2} [{(- 1)}^{n} Q_{n} - 1] \\ {(- 1)}^{n + 1} P_{n} & 0 & {(- 1)}^{n} P_{n + 1} \end{matrix})$

$= (\begin{matrix} x & 0 & y \\ z & 1 & t \\ k & 0 & l \end{matrix})$

is obtained, where

x = (-1) ^m+n [P_n-1P_m-1 + P_nP_m],

y = (-1) ^m+n+1 [P_n-1P_m + P_m+1P_n] ,

$z = \frac{k}{2} {(- 1)}^{m + n + 1} [Q_{n - 1} P_{m - 1} + - Q_{n} P_{m}] - \frac{k}{2},$

$t = \frac{k}{2} {(- 1)}^{m + n} [Q_{n - 1} P_{m} + Q_{n} P_{m + 1} - \frac{k}{2}$ ,

k = (-1) ^m+n+1 [P_nP_m-1 + P_n+1P_m]

l = (-1) ^m+n [P_nP_m + P_n+1P_m+1].

So, From the equation Z^m+n = Z^mZⁿ, we get the identities in the theorem.

Theorem 4. The following equalities are provided for $n, m \in ℤ^{+}$ ,

$\det (V_{L}^{n}) = {\begin{matrix} 2^{3 n} & forevenn, \\ - 2^{3 n} & foroddn, \end{matrix}$

P_2n² - P_2n+2P_2n-2 = 4,

Q_2n² - Q_2n+2Q_2n-2 = -8,

where P_n and Q_n are n^th Pell and Pell-Lucas numbers, respectively.

Proof. Firstly, we establish (i), by using mathematical induction on odd n. It is clear that, det (V_L) = -2³ for n = 1. Let us suppose $\det (V_{L}^{k}) = - 2^{3 k}$ for n = k. We must show the statement is true for n = k + 2. If we use the multiplicative property of the determinant, we get

$\begin{matrix} \det (V_{L}^{k + 2}) & = & \det (V_{L}^{k}) \det (V_{L}^{1}) \det (V_{L}^{1}) \\ = & - 2^{3 (k + 2)} . \end{matrix}$ For even values of n, the same motivation is used. So, (i) is proven for all $n \in ℤ^{+} .$ To prove identities (ii) and (iii), we can use equations (7) and (i) for even and odd n, respectively.

Theorem 5. The following equalities are provided for $n, m \in ℤ^{+}$ ;

4P_2m+2n = Q_2mQ_2n+2 - Q_2m-2Q_2n,

2P_2m+2n = P_2mP_2n+2 - P_2m-2P_2n,

2Q_2m+2n = Q_2mP_2n+2 - Q_2m-2P_2n,

2P_2m-2n = P_2m-2P_2n - P_2mP_2n-2,

4P_2m-2n = Q_2m-2Q_2n - Q_2mQ_2n-2,

2Q_2m-2n = P_2m-2Q_2n - P_2mQ_2n-2 .

Proof. By using equation (7), we can write matrix $V_{L}^{m + n}$ as

$V_{L}^{m + n} =$ ${\begin{matrix} 2^{\frac{3}{2} (m + n) - 1} [\begin{matrix} - P_{2 m + 2 n - 2} & P_{2 m + 2 n} \\ - P_{2 m + 2 n} & P_{2 m + 2 n + 2} \end{matrix}] foreven m+n, \\ 2^{\frac{3}{2} (m + n - 1)} [\begin{matrix} Q_{2 m + 2 n - 2} & - Q_{2 m + 2 n} \\ Q_{2 m + 2 n} & - Q_{2 m + 2 n + 2} \end{matrix}] forodd m+n . \end{matrix}$ (8)

Firstly, we examine for odd values m and n. So, we can write the following matrix as

$V_{L}^{m} V_{L}^{n} = 2^{\frac{3}{2} (m + n) - 1} 2^{- 2} \times$ $[\begin{matrix} Q_{2 m - 2} Q_{2 n - 2} - Q_{2 m} Q_{2 n} & Q_{2 m} Q_{2 n + 2} - Q_{2 m - 2} Q_{2 n} \\ Q_{2 m} Q_{2 n - 2} - Q_{2 m + 2} Q_{2 n} & Q_{2 m + 2} Q_{2 n + 2} - Q_{2 m} Q_{2 n} \end{matrix}] .$ (9)

If we compare equation (8) with equation (9), we get the following identity,

4 P_{2 m + 2 n} = Q_{2 m} Q_{2 n + 2} - Q_{2 m - 2} Q_{2 n} .

When we take even values m and n, we obtain the matrix as

$V_{L}^{m} V_{L}^{n} = 2^{\frac{3}{2} (m + n) - 1} 2^{- 1} \times$ $[\begin{matrix} P_{2 m - 2} P_{2 n - 2} - P_{2 m} Q_{Pn} & P_{2 m} P_{2 n + 2} - P_{2 m - 2} P_{2 n} \\ P_{2 m} P_{2 n - 2} - P_{2 m + 2} P_{2 n} & P_{2 m + 2} P_{2 n + 2} - P_{2 m} P_{2 n} \end{matrix}] .$ (10) By comparing equation (8) with equation (10), we find a new identity as $2 P_{2 m + 2 n} = P_{2 m} P_{2 n + 2} - P_{2 m - 2} P_{2 n} .$

Now, let us examine for even m and odd n or odd m and even n. So, we get the matrix as

$V_{L}^{m} V_{L}^{n} =$ $2^{\frac{3}{2} (m + n) - 1} 2^{- 1} \times$ $[\begin{matrix} Q_{2 m} P_{2 n} - Q_{2 m - 2} P_{2 n - 2} & Q_{2 m - 2} P_{2 n} - Q_{2 m} P_{2 n + 2} \\ Q_{2 m} P_{2 n - 2} - P_{2 n} Q_{2 m + 2} & Q_{2 m + 2} P_{2 n + 2} - Q_{2 m} P_{2 n} \end{matrix}] .$ (11)

When we compare equation (8) with equation (11), we obtain the following identity, $2 Q_{2 m + 2 n} = Q_{2 m} P_{2 n + 2} - Q_{2 m - 2} P_{2 n} .$ Now, let us take the inverse of the matrix $V_{L}^{n}$ , then $V_{L}^{- n} = {\begin{matrix} \frac{1}{2^{\frac{3}{2} n + 1}} [\begin{matrix} P_{2 n + 2} & - P_{2 n} \\ P_{2 n} & - P_{2 n - 2} \end{matrix}] foreven n, \\ \frac{1}{2^{\frac{3}{2} (n + 1)}} [\begin{matrix} - Q_{2 n + 2} & Q_{2 n} \\ - Q_{2 n} & Q_{2 n - 2} \end{matrix}] forodd n . \end{matrix}$ When we compute the equality $V_{L}^{m - n} = V_{L}^{m} V_{L}^{- n}$ , by using same motivation as above, we obtain following identities. For even values m and n, we get $2 P_{2 m - 2 n} = P_{2 m - 2} P_{2 n} - P_{2 m} P_{2 n - 2} .$ In the case of odd values m and n, we get $4 P_{2 m - 2 n} = Q_{2 m - 2} Q_{2 n} - Q_{2 m} Q_{2 n - 2} .$ Finally, if we take even m odd n or odd m and even n, we obtain $2 Q_{2 m - 2 n} = P_{2 m - 2} Q_{2 n} - P_{2 m} Q_{2 n - 2} .$

Now, by using the λ-function from [14], some identities which related to Pell and Pell-Lucas numbers are produced.

Theorem 6. If for all n ≥ 1, P_n is the n^th Pell number and Q_n is the n^th Pell- Lucas number, then

$Q_{n} Q_{n + 1} - 2 P_{n} P_{n + 1} = {(- 1)}^{n} .$

Proof. For the proof we use λ- function for matrix Zⁿ. Firstly, let us write function λ (Zⁿ) as

λ (Zⁿ)

$= | \begin{matrix} (- 1)^{n} [P_{n - 1} + \frac{k}{2} Q_{n - 1}] + \frac{k}{2} + 1 & (- 1)^{n} [P_{n} + \frac{k}{2} Q_{n}] - \frac{k}{2} - 1 \\ (- 1)^{n + 1} [- P_{n} + \frac{k}{2} Q_{n - 1}] - \frac{k}{2} - 1 & (- 1)^{n} [P_{n + 1} - \frac{k}{2} Q_{n}] + \frac{k}{2} + 1 \end{matrix} | .$

So, by using identities (1), (6) and (4), we obtain

$λ (Z^{n}) = (- 1)^{n} [1 - k + (k + 2) P_{n + 1}] .$ For the equation |Zⁿ^*| = |Zⁿ| + lλ (Zⁿ), we can find the matrix Zⁿ^* by choosing l = (-1) ⁿP_n. Then the matrix Zⁿ^* is obtained as follows:

Zⁿ^* =

$(\begin{matrix} (- 1)^{n} P_{n - 1} + (- 1)^{n} P_{n} & (- 1)^{n} P_{n} & 0 \\ (- 1)^{n} [P_{n} - \frac{k}{2} Q_{n - 1}] - \frac{k}{2} & 1 + (- 1)^{n} P_{n} & (- 1)^{n} P_{n} + \frac{k}{2} (- 1)^{n} Q_{n} - \frac{k}{2} \\ 0 & (- 1)^{n} P_{n} & (- 1)^{n} P_{n} + (- 1)^{n} P_{n + 1} ∥ \end{matrix}) .$ By using recurrence relations of Pell, Pell-Lucas numbers and identities (6) and (1), we get |Zⁿ^*| = Q_nQ_n+1 + (-1) ⁿP_n [Q_nQ_n+1 - 2P_nP_n+1] - kP_n + kP_nP_n+1 .

If we write these calculations in the equation |Zⁿ^*| = |Zⁿ| + lλ (Zⁿ), then we get Q_nQ_n+1 - 2P_nP_n+1 = (-1) ⁿ .

Theorem 7. If for all n ≥ 1, let P_n is the n^th Pell number and Q_n is the n^th Pell- Lucas number, then

$2 P_{2 n - 1} Q_{2 n + 1} = 2 + P_{4 n} .$

Proof. By using λ function for matrix $V_{L}^{n}$ ; firstly for even n, we get $λ (V_{L}^{n}) = - 2^{\frac{3}{2} n - 1} P_{2 n - 2} + 2^{\frac{3}{2} n - 1} P_{2 n + 2}$ (12) $\begin{matrix} - 2^{\frac{3}{2} n - 1} P_{2 n} + + 2^{\frac{3}{2} n - 1} P_{2 n} \\ = & 2^{\frac{3}{2} n - 1} (P_{2 n + 2} - P_{2 n - 2}) = 2^{\frac{3}{2} n - 1} 2^{2} Q_{2 n} \\ = & 2^{\frac{3}{2} n + 1} Q_{2 n} \end{matrix}$ (13) by using identity (2) and $| (V_{L}^{n})^{*} | = | V_{L}^{n} | + k λ (V_{L}^{n})$ , where $k = 2^{\frac{3}{2} n - 1} P_{2 n}$ and

${(V_{L}^{n})}^{*} = (\begin{matrix} 2^{\frac{3}{2} n - 1} (P_{2 n} - P_{2 n - 2}) & 2^{\frac{3}{2} n - 1} 2 P_{2 n} \\ 0 & 2^{\frac{3}{2} n - 1} (P_{2 n + 2} + P_{2 n}) \end{matrix})$ .

The determinant of matrix $V_{L}^{n}$ is $| V_{L}^{n} | = 2^{3 n}$ and the determinant of matrix ${V_{L}^{n}}^{*}$ is $| {V_{L}^{n}}^{*} | = 2^{2 (\frac{3}{2} n - 1)} (P_{2 n} - P_{2 n - 2}) (P_{2 n + 2} + P_{2 n})$ Therefore, $| {V_{L}^{n}}^{*} | = 2^{\frac{3}{2} n - 1} P_{2 n - 1}$ . If we write them in the equation $| (V_{L}^{n})^{*} | = | V_{L}^{n} | + k λ (V_{L}^{n})$ , then we get

$2^{2 (\frac{3}{2} n - 1)} (P_{2 n} - P_{2 n - 2}) (P_{2 n + 2} + P_{2 n}) =$

$2^{3 n} + (2^{\frac{3}{2} n - 1} P_{2 n}) (2^{\frac{3}{2} n + 1} Q_{2 n})$ .By using identities (2) and (3), that is 2P_2n-1Q_2n+1 = 2 + P_4n.

When we examine for odd n, the same identity is obtained.

References

Dasdemir

On the Pell, Pell-Lucas and modified Pell numbers bymatrix method,–, Applied Mathematical Sciences 5(64) (2011), 3173–3181.

Horadam

A.F.

Applications of modified pell numbers torepresentations, Ulam Quarterly 3 (1994), 34–53.

Horadam

A.F.

Pell identities, Fibonacci Quarterly 9(3) (1971), 245–263.

Değer

A.H.

Vertices of paths of minimal lengths on suborbitalgraphs, 31(4) (2017), 913–923.

Kilic

, Tasci

The linear algebra of the Pell matrix, Bolet&n de la Sociedad Matemática Mexicana 11(2005).

Köken

, Bozkurt

On Lucas numbers by the matrix method, Hacettepe Journal of Mathematics and Statistics 39(2010), 471–475.

Silvester

J.R.

Fibonacci properties by matrix methods, TheMathematical Gazette 63(425) (1979), 188–191.

Ercolano

Matrix generator of Pell sequence, FibonacciQuarterly 17(1) (1979), 71–77.

Keskin

, Demirtřk

Some new Fibonacci and Lucasidentities by matrix methods, International Journal of Mathematical Education in Science and Technology 41(3) (2010), 379–387.

10.

Koshy

Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.

11.

Aiyub

, Esi

, Subramanian

Poisson Fibonacci binomialmatrix on rough statistical convergence on triple sequences and itsrate, Journal of Intelligent & Fuzzy Systems 36(4) (2019), 3439–3445.

12.

TaŞ

, Uçar

, Yılmaz Özgür

Pell coding and pell decoding methods with some applications, arXiv preprint arXiv:1706.04377, (2017).

13.

Akbaba

Ü.

, Değer

A.H.

Relation between matrices and thesuborbital graphs by the special number sequences, TurkishJournal of Mathematics 42 (2022), 753–767.

14.

Hoggatt

V.E.

, Bicknell

Fibonacci matrices and lambdafunction, Fibonacci Quarterly 1 (1963), 47–52.

15.

Jiang

, Li

, Shen

On explicit determinants of RFPLR and RFPLL circulant matrices involving Pell numbers in information theory, International Conference on Information Computing and Applications, Springer, Berlin, Heidelberg, 2012. pp. 364–370.