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

Emrah KILIÇ Neşe ÖMÜR Sibel KOPARAL

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.

Anahtar Kelimeler

Generalized Filbert matrix, q-analogues, LU-decomposition, Zeilberger’s algorithm, Computer algebra system (CAS)

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.