Yıl 2020, Cilt 49 , Sayı 2, Sayfalar 684 - 694 2020-04-02

New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries

Emrah KILIÇ [1] , Neşe ÖMÜR [2] , Sibel KOPARAL [3]


In this paper, we present new analogues of the Filbert and Lilbert matrices via products of two $k$-tuples asymmetric entries consist of the Fibonacci and Lucas numbers. We shall derive explicit formulae for their $LU$-decompositions and inverses. To prove the claimed results, we write all the identities to be proven in $q$-word and then use the celebrated Zeilberger algorithm to prove required $q$-identities.
Generalized Filbert matrix, q-analogues, LU-decomposition, Zeilberger’s algorithm, Computer algebra system (CAS)
  • [1] C. Berg, Fibonacci numbers and orthogonal polynomials, Arab. J. Math. Sci. 17, 75– 88, 2011.
  • [2] L. Carlitz, Some determinants of q-binomial coefficients, J. Reine Angew. Math. 226, 216–220, 1967.
  • [3] W. Chu, On the evaluation of some determinants with q-binomial coefficients, J. Systems Sci. Math. Science 8 (4), 361–366, 1988.
  • [4] W. Chu, Generalizations of the Cauchy determinant, Publ. Math. Debrecen 58 (3), 353–365, 2001.
  • [5] W. Chu and L. Di Claudio, Binomial determinant evaluations, Ann. Comb. 9 (4), 363–377, 2005.
  • [6] W. Chu, Finite differences and determinant identities, Linear Algebra Appl. 430, 215–228, 2009.
  • [7] M.E.H. Ismail, One parameter generalizations of the Fibonacci and Lucas numbers, The Fibonacci Quart. 46/47, 167–180, 2008/2009.
  • [8] E. Kılıç and H. Prodinger, A generalized Filbert matrix, The Fibonacci Quart. 48, 29–33, 2010.
  • [9] E. Kılıç and H. Prodinger, The q-Pilbert matrix, Int. J. Comput. Math. 89, 1370– 1377, 2012.
  • [10] E. Kılıç and H. Prodinger, Variants of the Filbert matrix, The Fibonacci Quart. 51, 153–162, 2013.
  • [11] E. Kılıç and H. Prodinger, Asymmetric generalizations of the Filbert matrix and variants, Publ. Inst. Math. (Belgrad) (N.S) 95 (109), 267–280, 2014.
  • [12] E. Kılıç and H. Prodinger, The generalized q-Pilbert matrix, Math. Slovaca 64, 1083– 1092, 2014.
  • [13] E. Kılıç and H. Prodinger, The generalized Lilbert matrix, Periodica Math. Hungar. 73, 62–72, 2016.
  • [14] G.Y. Lee, S.G. Lee, and H.G. Shin, On the k-generalized Fibonacci matrix $Q_{K}^{\ast }$, Linear Algebra Appl. 251, 73–88, 1997.
  • [15] G.Y. Lee and S.H. Cho, The generalized Pascal matrix via the generalized Fibonacci matrix and the generalized Pell matrix, J. Korean Math. Soc. 45 (2), 479–491, 2008.
  • [16] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices 1, 10–16, 2013.
  • [17] A.M. Ostrowski, On some determinants with combinatorial numbers, J. Reine Angew. Math. 216, 25–30, 1964.
  • [18] M. Petkovsek, H. Wilf, and D. Zeilberger, A=B, A.K. Peters, Wellesley, MA, 1996.
  • [19] H. Prodinger, A generalization of a Filbert matrix with 3 additional parameters, Trans. Roy. Soc. South Afr. 65, 169–172, 2010.
  • [20] T.M. Richardson, The Filbert matrix, The Fibonacci Quart. 39 (3), 268–275, 2001.
  • [21] J. Zhou and J. Zhaolin, The spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput. 233, 582–587, 2014.
  • [22] J. Zhou and J. Zhaolin, Spectral norms of circulant-type matrices with binomial coefficients and Harmonic numbers, Int. J. Comput. Math. 11 (5), 1350076, 2014.
Birincil Dil en
Konular Matematik
Bölüm Matematik
Yazarlar

Orcid: 0000-0003-0722-7382
Yazar: Emrah KILIÇ
Kurum: TOBB University of Economics and Technology
Ülke: Turkey


Orcid: 0000-0002-3972-9910
Yazar: Neşe ÖMÜR
Kurum: Kocaeli University
Ülke: Turkey


Orcid: 0000-0001-9574-9652
Yazar: Sibel KOPARAL (Sorumlu Yazar)
Kurum: Kocaeli University
Ülke: Turkey


Tarihler

Yayımlanma Tarihi : 2 Nisan 2020

Bibtex @araştırma makalesi { hujms473495, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2020}, volume = {49}, pages = {684 - 694}, doi = {10.15672/hujms.473495}, title = {New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries}, key = {cite}, author = {KILIÇ, Emrah and ÖMÜR, Neşe and KOPARAL, Sibel} }
APA KILIÇ, E , ÖMÜR, N , KOPARAL, S . (2020). New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries. Hacettepe Journal of Mathematics and Statistics , 49 (2) , 684-694 . DOI: 10.15672/hujms.473495
MLA KILIÇ, E , ÖMÜR, N , KOPARAL, S . "New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 684-694 <https://dergipark.org.tr/tr/pub/hujms/issue/53568/473495>
Chicago KILIÇ, E , ÖMÜR, N , KOPARAL, S . "New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 684-694
RIS TY - JOUR T1 - New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries AU - Emrah KILIÇ , Neşe ÖMÜR , Sibel KOPARAL Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.473495 DO - 10.15672/hujms.473495 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 684 EP - 694 VL - 49 IS - 2 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.473495 UR - https://doi.org/10.15672/hujms.473495 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries %A Emrah KILIÇ , Neşe ÖMÜR , Sibel KOPARAL %T New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 2 %R doi: 10.15672/hujms.473495 %U 10.15672/hujms.473495
ISNAD KILIÇ, Emrah , ÖMÜR, Neşe , KOPARAL, Sibel . "New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries". Hacettepe Journal of Mathematics and Statistics 49 / 2 (Nisan 2020): 684-694 . https://doi.org/10.15672/hujms.473495
AMA KILIÇ E , ÖMÜR N , KOPARAL S . New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 684-694.
Vancouver KILIÇ E , ÖMÜR N , KOPARAL S . New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 694-684.