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Explicit Determinants of the RFPLR Circulant and RLPFL Circulant Matrices Involving Some Famous Numbers
Circulant matrices may play a crucial role in solving various differential equations. In this paper, the techniques used herein are based on the inverse factorization of polynomial. We give the explicit determinants of the RFPLR circulant matrices and RLPFL circulant matrices involving Fibonacci, Lucas, Pell, and Pell-Lucas number, respectively.
It has been found out that circulant matrices play an important role in solving differential equations in various fields such as Lin and Yang discretized the partial integrodifferential equation (PIDE) in pricing options with the preconditioned conjugate gradient (PCG) method, where constructed the circulant preconditioners. By using the FFT, the cost for each linear system is where is the size of the system in . Lei and Sun  proposed the preconditioned CGNR (PCGNR) method with a circulant preconditioner to solve such Toeplitz-like systems. Kloeden et al. adopted the simplest approximation schemes for (1) in  with the Euler method, which reads (5) in . They exploited that the covariance matrix of the increments can be embedded in a circulant matrix. The total loops can be done by fast Fourier transformation, which leads to a total computational cost of . By using a Strang-type block-circulant preconditioner, Zhang et al.  speeded up the convergent rate of boundary-value methods. In , the resulting dense linear system exhibits so much structure that it can be solved very efficiently by a circulant preconditioned conjugate gradient method. Ahmed et al. used coupled map lattices (CML) as an alternative approach to include spatial effects in FOS. Consider the 1-system CML (10) in . They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy (2) in . Wu and Zou in  discussed the existence and approximation of solutions of asymptotic or periodic boundary-value problems of mixed functional differential equations. They focused on (5.13) in  with a circulant matrix, whose principal diagonal entries are zeroes.
Circulant matrix family have important applications in various disciplines including image processing, communications, signal processing, encoding, and preconditioner. They have been put on firm basis with the work of Davis  and Jiang and Zhou . The circulant matrices, long a fruitful subject of research, have in recent years been extended in many directions [10–13]. The -circulant matrices are another natural extension of this well-studied class and can be found in [14–20]. The -circulant matrix has a wide application, especially on the generalized cyclic codes in . The properties and structures of the -circulant matrices, which are called RFPLR circulant matrices, are better than those of the general -circulant matrices, so there are good algorithms for determinants.
There are many interests in properties and generalization of some special matrices with famous numbers. Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers . Dazheng gave the determinant of the Fibonacci-Lucas quasicyclic matrices . Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers in . Shen et al.  discussed the determinant of circulant matrix involving Fibonacci and Lucas numbers. Akbulak and Bozkurt  gave the norms of Toeplitz involving Fibonacci and Lucas numbers. The authors [26, 27] discussed some properties of Fibonacci and Lucas matrices. Stanimirović et al. gave generalized Fibonacci and Lucas matrix in . Z. Zhang and Y. Zhang  investigated the Lucas matrix and some combinatorial identities.
Firstly, we introduce the definitions of the RFPLR circulant matrices and RLPFL circulant matrices and properties of the related famous numbers. Then, we present the main results and the detailed process.
2. Definition and Lemma
Definition 1. A row first-plus-last -right (RFPLR) circulant matrix with the first row , denoted by , means a square matrix of the form
Note that the RFPLR circulant matrix is a circulant matrix, which is neither an extension nor special case of the circulant matrix . They are two completely different kinds of special matrices.
We define as the basic RFPLR circulant matrix; that is, Both the minimal polynomial and the characteristic polynomial of are , which has only simple roots, denoted by . In addition, satisfies and . Then a matrix can be written in the form if and only if is a RFPLR circulant matrix, where the polynomial is called the representer of the RFPLR circulant matrix .
Since is nonderogatory, then is a RFMLR circulant matrix if and only if commutes with ; that is, . Because of the representation, RFMLR circulant matrices have very nice structure and the algebraic properties also can be easily attained. Moreover, the product of two RFMLR circulant matrices and the inverse are again RFMLR circulant matrices.
Definition 2. A row last-plus-first -left (RLPFL) circulant matrix with the first row , denoted by , means a square matrix of the form
Let and . By explicit computation, we find where is the backward identity matrix of the form
The first few values of these sequences are given by the following table ():
The sequences , , , and are given by the Binet formulae where , are the roots of the characteristic equation and , are the roots of the characteristic equation .
By Proposition 5.1 in , we deduce the following lemma.
Lemma 3. Let ; then the eigenvalues of are and in addition, where are the roots of the equation
Lemma 4. Consider where and satisfy (12), , .
Since satisfy (12), we must have So
3. Determinant of the RFPLR and RLPFL Circulant Matrices with the Fibonacci Numbers
Theorem 5. Let . Then where
Using the method in Theorem 5 similarly, we also have the following.
Theorem 6. Let . Then
Theorem 7. Let . Then
Proof. The matrix can be written as
Hence, we have
where and its determinant is obtained from Theorem 6,
In addition, so
4. Determinant of the RFMLR and RLMFL Circulant Matrices with the Lucas Numbers
Theorem 8. Let . Then where
Using the method in Theorem 8 similarly, we also have the following.
Theorem 9. Let . Then where
Theorem 10. Let . Then where
Proof. The matrix can be written as
Thus, we have where matrix and its determinant can be obtained from Theorem 9, where
In addition, so the determinant of matrix is where
5. Determinants of the RFPLR and RLPFL Circulant Matrix with the Pell Numbers
Theorem 11. If , then where
Using the method in Theorem 11 similarly, we also have the following.
Theorem 12. If , then
Theorem 13. If , then one has
Proof. The matrix can be written as Then we can get where and its determinant could be obtained through Theorem 12; namely, So
6. Determinants of the RFPLR and RLPFL Circulant Matrix with the Pell-Lucas Numbers
Theorem 14. If , then one has where
Proof. The method is similar to Theorem 11.
Certainly, we can get the following theorem.
Theorem 15. If , then one gets where
Theorem 16. If , then where
The determinant problems of the RFPLR circulant matrices and RLPFL circulant matrices involving the Fibonacci, Lucas, Pell, and Pell-Lucas number are considered in this paper. The explicit determinants are presented by using some terms of these numbers.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11301252) and the AMEP of Linyi University, China.
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