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

Yıl 2020, Cilt: 49 Sayı: 2, 684 - 694, 02.04.2020
https://doi.org/10.15672/hujms.473495

Öz

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.

Kaynakça

  • [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.
Yıl 2020, Cilt: 49 Sayı: 2, 684 - 694, 02.04.2020
https://doi.org/10.15672/hujms.473495

Öz

Kaynakça

  • [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.
Toplam 22 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Emrah Kılıç 0000-0003-0722-7382

Neşe Ömür Bu kişi benim 0000-0002-3972-9910

Sibel Koparal 0000-0001-9574-9652

Yayımlanma Tarihi 2 Nisan 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 49 Sayı: 2

Kaynak Göster

APA Kılıç, 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. https://doi.org/10.15672/hujms.473495
AMA Kılıç 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. Nisan 2020;49(2):684-694. doi:10.15672/hujms.473495
Chicago Kılıç, Emrah, Neşe Ömür, ve Sibel Koparal. “New Analogues of the Filbert and Lilbert Matrices via Products of Two K-Tuples Asymmetric Entries”. Hacettepe Journal of Mathematics and Statistics 49, sy. 2 (Nisan 2020): 684-94. https://doi.org/10.15672/hujms.473495.
EndNote Kılıç E, Ömür N, Koparal S (01 Nisan 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.
IEEE E. Kılıç, N. Ömür, ve S. Koparal, “New analogues of the Filbert and Lilbert matrices via products of two k-tuples asymmetric entries”, Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 2, ss. 684–694, 2020, doi: 10.15672/hujms.473495.
ISNAD Kılıç, Emrah vd. “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.
JAMA Kılıç 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;49:684–694.
MLA Kılıç, Emrah vd. “New Analogues of the Filbert and Lilbert Matrices via Products of Two K-Tuples Asymmetric Entries”. Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 2, 2020, ss. 684-9, doi:10.15672/hujms.473495.
Vancouver Kılıç 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):684-9.