## Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales

View this Special IssueResearch Article | Open Access

# Explicit Determinants of the RFPLR Circulant and RLPFL Circulant Matrices Involving Some Famous Numbers

**Academic Editor:**Tongxing Li

#### Abstract

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.

#### 1. Introduction

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 [1]. Lei and Sun [2] 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 [3] with the Euler method, which reads (5) in [3]. 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. [4] speeded up the convergent rate of boundary-value methods. In [5], 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 [6]. They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy (2) in [6]. Wu and Zou in [7] 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 [7] 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 [8] and Jiang and Zhou [9]. 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 [14]. 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 [21]. Dazheng gave the determinant of the Fibonacci-Lucas quasicyclic matrices [22]. Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers in [23]. Shen et al. [24] discussed the determinant of circulant matrix involving Fibonacci and Lucas numbers. Akbulak and Bozkurt [25] 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 [28]. Z. Zhang and Y. Zhang [29] 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 [8]. 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 Fibonacci, Lucas, Pell, and the Pell-Lucas sequences [30–36] are defined by the following recurrence relations, respectively:

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 [14], 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), , .*

*Proof. *Consider
while

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
*

*Proof. *The matrix can be written as
Using Lemma 3, the determinant of is
Using Lemma 4, we obtain
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
*

*Proof. *The matrix can be written as

Using Lemma 3, we have

According to Lemma 4, we obtain

Then, we get
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
*

*Proof. *The matrix can be written as

Using Lemma 3, the determinant of is

According to Lemma 4, we can get
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
*

#### 7. Conclusion

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.

#### Acknowledgments

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.

#### References

- F.-R. Lin and H.-X. Yang, “A fast stationary iterative method for a partial integro-differential equation in pricing options,”
*Calcolo*, vol. 50, no. 4, pp. 313–327, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S.-L. Lei and H.-W. Sun, “A circulant preconditioner for fractional diffusion equations,”
*Journal of Computational Physics*, vol. 242, pp. 715–725, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. E. Kloeden, A. Neuenkirch, and R. Pavani, “Multilevel Monte Carlo for stochastic differential equations with additive fractional noise,”
*Annals of Operations Research*, vol. 189, pp. 255–276, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Zhang, H. Chen, and L. Wang, “Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations,”
*Numerical Linear Algebra with Applications*, vol. 18, no. 5, pp. 843–855, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. W. Sachs and A. K. Strauss, “Efficient solution of a partial integro-differential equation in finance,”
*Applied Numerical Mathematics. An IMACS Journal*, vol. 58, no. 11, pp. 1687–1703, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,”
*Physica A: Statistical Mechanics and its Applications*, vol. 379, no. 2, pp. 607–614, 2007. View at: Publisher Site | Google Scholar - J. Wu and X. Zou, “Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,”
*Journal of Differential Equations*, vol. 135, no. 2, pp. 315–357, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. J. Davis,
*Circulant Matrices*, John Wiley & Sons, New York, NY, USA, 1979. View at: MathSciNet - Z. L. Jiang and Z. X. Zhou,
*Circulant Matrices*, Chengdu Technology University, Chengdu, China, 1999. - Z. L. Jiang, “On the minimal polynomials and the inverses of multilevel scaled factor circulant matrices,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 521643, 10 pages, 2014. View at: Publisher Site | Google Scholar - Z. Jiang, T. Xu, and F. Lu, “Isomorphic operators and functional equations for the skew-circulant algebra,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 418194, 8 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Z. Jiang, Y. Gong, and Y. Gao, “Invertibility and explicit inverses of circulant-type matrices with $k$-Fibonacci and $k$-Lucas numbers,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 238953, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - J. Li, Z. Jiang, and F. Lu, “Determinants, norms, and the spread of circulant matrices with tribonacci and generalized Lucas numbers,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 381829, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - D. Chillag, “Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes,”
*Linear Algebra and its Applications*, vol. 218, pp. 147–183, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-L. Jiang and Z.-B. Xu, “Efficient algorithm for finding the inverse and the group inverse of FLS $r$-circulant matrix,”
*Journal of Applied Mathematics & Computing*, vol. 18, no. 1-2, pp. 45–57, 2005. View at: Publisher Site | Google Scholar | MathSciNet - Z. L. Jiang and D. H. Sun, “Fast algorithms for solving the inverse problem of Ax = b,” in
*Proceedings of the 8th International Conference on Matrix Theory and Its Applications in China*, pp. 121C–124C, 2008. View at: Google Scholar - J. Li, Z. Jiang, and N. Shen, “Explicit determinants of the Fibonacci RFPLR circulant and Lucas RFPLR circulant matrix,”
*JP Journal of Algebra, Number Theory and Applications*, vol. 28, no. 2, pp. 167–179, 2013. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z. P. Tian, “Fast algorithm for solving the first-plus last-circulant linear system,”
*Journal of Shandong University: Natural Science*, vol. 46, no. 12, pp. 96–103, 2011. View at: Google Scholar | MathSciNet - N. Shen, Z. L. Jiang, and J. Li, “On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers,”
*WSEAS Transactions on Mathematics*, vol. 12, no. 1, pp. 42–53, 2013. View at: Google Scholar - Z. Tian, “Fast algorithms for solving the inverse problem of $AX=b$ in four different families of patterned matrices,”
*Far East Journal of Applied Mathematics*, vol. 52, no. 1, pp. 1–12, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - D. V. Jaiswal, “On determinants involving generalized Fibonacci numbers,”
*The Fibonacci Quarterly: Official Organ of the Fibonacci Association*, vol. 7, pp. 319–330, 1969. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Dazheng, “Fibonacci-Lucas quasi-cyclic matrices,”
*The Fibonacci Quarterly: The Official Journal of the Fibonacci Association*, vol. 40, no. 3, pp. 280–286, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - D. A. Lind, “A Fibonacci circulant,”
*The Fibonacci Quarterly: Official Organ of the Fibonacci Association*, vol. 8, no. 5, pp. 449–455, 1970. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S.-Q. Shen, J.-M. Cen, and Y. Hao, “On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers,”
*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9790–9797, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Akbulak and D. Bozkurt, “On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers,”
*Hacettepe Journal of Mathematics and Statistics*, vol. 37, no. 2, pp. 89–95, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G.-Y. Lee, J.-S. Kim, and S.-G. Lee, “Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices,”
*The Fibonacci Quarterly: The Official Journal of the Fibonacci Association*, vol. 40, no. 3, pp. 203–211, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Miladinović and P. Stanimirović, “Singular case of generalized Fibonacci and Lucas matrices,”
*Journal of the Korean Mathematical Society*, vol. 48, no. 1, pp. 33–48, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Stanimirović, J. Nikolov, and I. Stanimirović, “A generalization of Fibonacci and Lucas matrices,”
*Discrete Applied Mathematics: The Journal of Combinatorial Algorithms, Informatics and Computational Sciences*, vol. 156, no. 14, pp. 2606–2619, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhang and Y. Zhang, “The Lucas matrix and some combinatorial identities,”
*Indian Journal of Pure and Applied Mathematics*, vol. 38, no. 5, pp. 457–465, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. Yilmaz and D. Bozkurt, “Hessenberg matrices and the Pell and Perrin numbers,”
*Journal of Number Theory*, vol. 131, no. 8, pp. 1390–1396, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Blümlein, D. J. Broadhurst, and J. A. M. Vermaseren, “The multiple zeta value data mine,”
*Computer Physics Communications*, vol. 181, no. 3, pp. 582–625, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Janjić, “Determinants and recurrence sequences,”
*Journal of Integer Sequences*, vol. 15, no. 3, pp. 1–2, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Elia, “Derived sequences, the Tribonacci recurrence and cubic forms,”
*The Fibonacci Quarterly: The Official Journal of the Fibonacci Association*, vol. 39, no. 2, pp. 107–115, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. Melham, “Sums involving Fibonacci and Pell numbers,”
*Portugaliae Mathematica*, vol. 56, no. 3, pp. 309–317, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Y. Yazlik and N. Taskara, “A note on generalized $k$-Horadam sequence,”
*Computers & Mathematics with Applications*, vol. 63, no. 1, pp. 36–41, 2012. View at: Publisher Site | Google Scholar | MathSciNet - E. Kılıç, “The generalized Pell $(p,i)$-numbers and their Binet formulas, combinatorial representations, sums,”
*Chaos, Solitons & Fractals*, vol. 40, no. 4, pp. 2047–2063, 2009. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Tingting Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.