Research ArticleExplicit Determinants of the RFP119903L119903R Circulant and RLP119903F119903LCirculant Matrices Involving Some Famous Numbers
Tingting Xu12 Zhaolin Jiang1 and Ziwu Jiang1
1 Department of Mathematics Linyi University Linyi Shandong 276005 China2 School of Mathematical Sciences Shandong Normal University Jinan 250014 China
Correspondence should be addressed to Zhaolin Jiang jzh1208sinacom
Received 28 April 2014 Accepted 5 June 2014 Published 19 June 2014
Academic Editor Tongxing Li
Copyright copy 2014 Tingting Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Circulant matrices may play a crucial role in solving various differential equations In this paper the techniques used herein arebased on the inverse factorization of polynomialWe give the explicit determinants of the RFP119903L119903R circulantmatrices and RLP119903F119903Lcirculant matrices involving Fibonacci Lucas Pell and Pell-Lucas number respectively
1 Introduction
It has been found out that circulant matrices play an impor-tant role in solving differential equations in various fields suchas Lin and Yang discretized the partial integrodifferentialequation (PIDE) in pricing options with the preconditionedconjugate gradient (PCG) method where constructed thecirculant preconditioners By using the FFT the cost foreach linear system is 119874(119899 log 119899) where 119899 is the size of thesystem in [1] Lei and Sun [2] proposed the preconditionedCGNR (PCGNR) method with a circulant preconditionerto solve such Toeplitz-like systems Kloeden et al adoptedthe simplest approximation schemes for (1) in [3] with theEuler method which reads (5) in [3] They exploited thatthe covariance matrix of the increments can be embeddedin a circulant matrix The total loops can be done by fastFourier transformation which leads to a total computationalcost of 119874(119898 log119898) = 119874(119899 log 119899) By using a Strang-typeblock-circulant preconditioner Zhang et al [4] speeded upthe convergent rate of boundary-value methods In [5] theresulting dense linear system exhibits so much structure thatit can be solved very efficiently by a circulant preconditionedconjugate gradient method Ahmed et al used coupled maplattices (CML) as an alternative approach to include spatialeffects in FOS Consider the 1-system CML (10) in [6] Theyclaimed that the system is stable if all the eigenvalues ofthe circulant matrix satisfy (2) in [6] Wu and Zou in [7]
discussed the existence and approximation of solutions ofasymptotic or periodic boundary-value problems of mixedfunctional differential equationsThey focused on (513) in [7]with a circulant matrix whose principal diagonal entries arezeroes
Circulant matrix family have important applications invarious disciplines including image processing communica-tions signal processing encoding and preconditioner Theyhave been put on firm basis with the work of Davis [8] andJiang and Zhou [9] The circulant matrices long a fruitfulsubject of research have in recent years been extended inmany directions [10ndash13] The 119891(119909)-circulant matrices areanother natural extension of this well-studied class and canbe found in [14ndash20] The 119891(119909)-circulant matrix has a wideapplication especially on the generalized cyclic codes in [14]The properties and structures of the 119909119899 minus 119903119909 minus 119903-circulantmatrices which are called RFP119903L119903R circulant matrices arebetter than those of the general 119891(119909)-circulant matrices sothere are good algorithms for determinants
There are many interests in properties and generalizationof some special matrices with famous numbers Jaiswalevaluated some determinants of circulant whose elementsare the generalized Fibonacci numbers [21] Dazheng gavethe determinant of the Fibonacci-Lucas quasicyclic matrices[22] Lind presented the determinants of circulant and skewcirculant involving Fibonacci numbers in [23] Shen et al[24] discussed the determinant of circulant matrix involving
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 647030 9 pageshttpdxdoiorg1011552014647030
2 Abstract and Applied Analysis
Fibonacci and Lucas numbers Akbulak and Bozkurt [25]gave the norms of Toeplitz involving Fibonacci and Lucasnumbers The authors [26 27] discussed some propertiesof Fibonacci and Lucas matrices Stanimirovic et al gavegeneralized Fibonacci and Lucas matrix in [28] Z Zhangand Y Zhang [29] investigated the Lucas matrix and somecombinatorial identities
Firstly we introduce the definitions of theRFP119903L119903Rcircu-lant matrices and RLP119903F119903L circulant matrices and propertiesof the related famous numbers Then we present the mainresults and the detailed process
2 Definition and Lemma
Definition 1 A row first-plus-119903last 119903-right (RFP119903L119903R) circu-lant matrix with the first row (119886
Note that the RFP119903L119903R circulant matrix is a 119909119899 minus 119903119909 minus 119903circulant matrix which is neither an extension nor specialcase of the circulant matrix [8] They are two completelydifferent kinds of special matrices
We define Θ(119903119903)
as the basic RFP119903L119903R circulant matrixthat is
if and only if 119860 is a RFP119903L119903R circulant matrix where thepolynomial 119891(119909) = sum
119899minus1
119894=0119886119894119909119894 is called the representer of the
RFP119903L119903R circulant matrix 119860Since Θ
(119903119903)is nonderogatory then 119860 is a RFM119903L119903R
circulant matrix if and only if119860 commutes withΘ(119903119903)
that is119860Θ(119903119903)
= Θ(119903119903)
119860 Because of the representation RFM119903L119903Rcirculant matrices have very nice structure and the algebraicproperties also can be easily attained Moreover the productof two RFM119903L119903R circulant matrices and the inverse 119860minus1 areagain RFM119903L119903R circulant matrices
Definition 2 A row last-plus-119903first 119903-left (RLP119903F119903L) circu-lant matrix with the first row (119886
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
Fibonacci and Lucas numbers Akbulak and Bozkurt [25]gave the norms of Toeplitz involving Fibonacci and Lucasnumbers The authors [26 27] discussed some propertiesof Fibonacci and Lucas matrices Stanimirovic et al gavegeneralized Fibonacci and Lucas matrix in [28] Z Zhangand Y Zhang [29] investigated the Lucas matrix and somecombinatorial identities
Firstly we introduce the definitions of theRFP119903L119903Rcircu-lant matrices and RLP119903F119903L circulant matrices and propertiesof the related famous numbers Then we present the mainresults and the detailed process
2 Definition and Lemma
Definition 1 A row first-plus-119903last 119903-right (RFP119903L119903R) circu-lant matrix with the first row (119886
Note that the RFP119903L119903R circulant matrix is a 119909119899 minus 119903119909 minus 119903circulant matrix which is neither an extension nor specialcase of the circulant matrix [8] They are two completelydifferent kinds of special matrices
We define Θ(119903119903)
as the basic RFP119903L119903R circulant matrixthat is
if and only if 119860 is a RFP119903L119903R circulant matrix where thepolynomial 119891(119909) = sum
119899minus1
119894=0119886119894119909119894 is called the representer of the
RFP119903L119903R circulant matrix 119860Since Θ
(119903119903)is nonderogatory then 119860 is a RFM119903L119903R
circulant matrix if and only if119860 commutes withΘ(119903119903)
that is119860Θ(119903119903)
= Θ(119903119903)
119860 Because of the representation RFM119903L119903Rcirculant matrices have very nice structure and the algebraicproperties also can be easily attained Moreover the productof two RFM119903L119903R circulant matrices and the inverse 119860minus1 areagain RFM119903L119903R circulant matrices
Definition 2 A row last-plus-119903first 119903-left (RLP119903F119903L) circu-lant matrix with the first row (119886
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
minus 119903119876119899+ 21199032(minus1)119899(119899minus1)2
(67)
where
119870 = (minus2119903 minus 119876119899minus1
) (119892119899minus1
6+ ℎ119899minus1
6) + 2119903 (119892
119899
6+ ℎ119899
6)
minus 119903 (119876119899minus 119876119899minus1
)
1198926=
119876119899+ radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
ℎ6=
119876119899minus radic119876
2
119899+ 8119903119876
119899minus1+ 161199032
4119903
(68)
7 Conclusion
Thedeterminant problems of the RFP119903L119903R circulantmatricesand RLP119903F119903L circulant matrices involving the FibonacciLucas Pell and Pell-Lucas number are considered in thispaperThe explicit determinants are presented by using someterms of these numbers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Development Project ofScience amp Technology of Shandong Province (Grant no2012GGX10115) and NSFC (Grant no 11301252) and theAMEP of Linyi University China
References
[1] F-R Lin and H-X Yang ldquoA fast stationary iterative methodfor a partial integro-differential equation in pricing optionsrdquoCalcolo vol 50 no 4 pp 313ndash327 2013
[2] S-L Lei and H-W Sun ldquoA circulant preconditioner for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 242 pp 715ndash725 2013
[3] P E Kloeden A Neuenkirch and R Pavani ldquoMultilevelMonte Carlo for stochastic differential equations with additivefractional noiserdquo Annals of Operations Research vol 189 pp255ndash276 2011
[4] C Zhang H Chen and L Wang ldquoStrang-type preconditionersapplied to ordinary and neutral differential-algebraic equa-tionsrdquo Numerical Linear Algebra with Applications vol 18 no5 pp 843ndash855 2011
[5] E W Sachs and A K Strauss ldquoEfficient solution of a partialintegro-differential equation in financerdquo Applied NumericalMathematics An IMACS Journal vol 58 no 11 pp 1687ndash17032008
[6] E Ahmed and A S Elgazzar ldquoOn fractional order differentialequations model for nonlocal epidemicsrdquo Physica A StatisticalMechanics and its Applications vol 379 no 2 pp 607ndash614 2007
[7] J Wu and X Zou ldquoAsymptotic and periodic boundary valueproblems of mixed FDEs and wave solutions of lattice differ-ential equationsrdquo Journal of Differential Equations vol 135 no2 pp 315ndash357 1997
[8] P J Davis Circulant Matrices John Wiley amp Sons New YorkNY USA 1979
[9] Z L Jiang and Z X Zhou Circulant Matrices ChengduTechnology University Chengdu China 1999
[10] Z L Jiang ldquoOn the minimal polynomials and the inversesof multilevel scaled factor circulant matricesrdquo Abstract andApplied Analysis vol 2014 Article ID 521643 10 pages 2014
[11] Z Jiang T Xu and F Lu ldquoIsomorphic operators and functionalequations for the skew-circulant algebrardquo Abstract and AppliedAnalysis vol 2014 Article ID 418194 8 pages 2014
[12] Z Jiang Y Gong and Y Gao ldquoInvertibility and explicitinverses of circulant-type matrices with 119896-Fibonacci and 119896-Lucas numbersrdquoAbstract andAppliedAnalysis vol 2014 ArticleID 238953 9 pages 2014
[13] J Li Z Jiang and F Lu ldquoDeterminants norms and the spreadof circulant matrices with tribonacci and generalized Lucasnumbersrdquo Abstract and Applied Analysis vol 2014 Article ID381829 9 pages 2014
[14] D Chillag ldquoRegular representations of semisimple algebrasseparable field extensions group characters generalized cir-culants and generalized cyclic codesrdquo Linear Algebra and itsApplications vol 218 pp 147ndash183 1995
[15] Z-L Jiang and Z-B Xu ldquoEfficient algorithm for finding theinverse and the group inverse of FLS 119903-circulantmatrixrdquo Journalof Applied Mathematics amp Computing vol 18 no 1-2 pp 45ndash572005
[16] Z L Jiang and D H Sun ldquoFast algorithms for solving theinverse problem of Ax = brdquo in Proceedings of the 8th Inter-national Conference on Matrix Theory and Its Applications inChina pp 121Cndash124C 2008
[17] J Li Z Jiang and N Shen ldquoExplicit determinants of theFibonacci RFPLR circulant and Lucas RFPLR circulant matrixrdquoJP Journal of Algebra Number Theory and Applications vol 28no 2 pp 167ndash179 2013
[18] Z P Tian ldquoFast algorithm for solving the first-plus last-circulantlinear systemrdquo Journal of Shandong University Natural Sciencevol 46 no 12 pp 96ndash103 2011
[19] N Shen Z L Jiang and J Li ldquoOn explicit determinants ofthe RFMLR and RLMFL circulant matrices involving certain
Abstract and Applied Analysis 9
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
famous numbersrdquoWSEAS Transactions onMathematics vol 12no 1 pp 42ndash53 2013
[20] Z Tian ldquoFast algorithms for solving the inverse problem of119860119883 = 119887 in four different families of patterned matricesrdquo FarEast Journal of AppliedMathematics vol 52 no 1 pp 1ndash12 2011
[21] D V Jaiswal ldquoOn determinants involving generalized Fibonaccinumbersrdquo The Fibonacci Quarterly Official Organ of theFibonacci Association vol 7 pp 319ndash330 1969
[22] L Dazheng ldquoFibonacci-Lucas quasi-cyclic matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 280ndash286 2002
[23] D A Lind ldquoA Fibonacci circulantrdquo The Fibonacci QuarterlyOfficial Organ of the Fibonacci Association vol 8 no 5 pp 449ndash455 1970
[24] S-Q Shen J-M Cen and Y Hao ldquoOn the determinantsand inverses of circulant matrices with Fibonacci and Lucasnumbersrdquo Applied Mathematics and Computation vol 217 no23 pp 9790ndash9797 2011
[25] MAkbulak andD Bozkurt ldquoOn the norms of Toeplitzmatricesinvolving Fibonacci and Lucas numbersrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 2 pp 89ndash95 2008
[26] G-Y Lee J-S Kim and S-G Lee ldquoFactorizations and eigen-values of Fibonacci and symmetric Fibonacci matricesrdquo TheFibonacci Quarterly The Official Journal of the Fibonacci Asso-ciation vol 40 no 3 pp 203ndash211 2002
[27] M Miladinovic and P Stanimirovic ldquoSingular case of gener-alized Fibonacci and Lucas matricesrdquo Journal of the KoreanMathematical Society vol 48 no 1 pp 33ndash48 2011
[28] P Stanimirovic J Nikolov and I Stanimirovic ldquoA generaliza-tion of Fibonacci and Lucas matricesrdquo Discrete Applied Math-ematics The Journal of Combinatorial Algorithms Informaticsand Computational Sciences vol 156 no 14 pp 2606ndash26192008
[29] Z Zhang and Y Zhang ldquoThe Lucas matrix and some combina-torial identitiesrdquo Indian Journal of Pure and Applied Mathemat-ics vol 38 no 5 pp 457ndash465 2007
[30] F Yilmaz and D Bozkurt ldquoHessenberg matrices and the Pelland Perrin numbersrdquo Journal of Number Theory vol 131 no 8pp 1390ndash1396 2011
[31] J Blumlein D J Broadhurst and J A M Vermaseren ldquoThemultiple zeta value data minerdquo Computer Physics Communica-tions vol 181 no 3 pp 582ndash625 2010
[32] M Janjic ldquoDeterminants and recurrence sequencesrdquo Journal ofInteger Sequences vol 15 no 3 pp 1ndash2 2012
[33] M Elia ldquoDerived sequences the Tribonacci recurrence andcubic formsrdquoTheFibonacci QuarterlyTheOfficial Journal of theFibonacci Association vol 39 no 2 pp 107ndash115 2001
[34] R Melham ldquoSums involving Fibonacci and Pell numbersrdquoPortugaliae Mathematica vol 56 no 3 pp 309ndash317 1999
[35] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012
[36] E Kılıc ldquoThe generalized Pell (119901 119894)-numbers and their Binetformulas combinatorial representations sumsrdquo Chaos Solitonsamp Fractals vol 40 no 4 pp 2047ndash2063 2009
![FACTORING MATRICES INTO THE PRODUCT OF CIRCULANT AND ...math.aalto.fi/~mhuhtane/manu.pdf · garding applications, monomial matrices appear in representation theory [5, 19] and in](https://static.documents.pub/doc/80x56/5f0b84427e708231d430e7c2/factoring-matrices-into-the-product-of-circulant-and-mathaaltofimhuhtanemanupdf.jpg)
![arXiv:1805.05533v2 [eess.SP] 24 Jan 2020 · A TUTORIAL ON CIRCULANT MATRICES, CIRCULAR CONVOLUTION, AND THE DISCRETE FOURIER TRANSFORM BASSAM BAMIEH Key words. Discrete Fourier Transform,](https://static.documents.pub/doc/80x56/5ea82f73d184a021b230cc6d/arxiv180505533v2-eesssp-24-jan-2020-a-tutorial-on-circulant-matrices-circular.jpg)
