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The generalized Fibonomial matrix. (English) Zbl 1184.15021
After defining the generalized Fibonomial matrix and deriving a linear recurrence relation for the generalized Fibonacci coefficients, the author shows that the generalized Fibonomial and Pascal matrices have the same characteristic polynomials i.e. the same eigenvalues. By using matrix methods, explicit and closed formulas for the coefficients and their sums are obtained. In this respect generating functions, properties and combinatorial representations are derived. Additionally, some relationships between determinants of certain matrices and the generalized Fibonacci coefficients are presented. Some applications are also given as illustrative examples.
MSC:
15B36Matrices of integers
11B39Fibonacci and Lucas numbers, etc.
05A19Combinatorial identities, bijective combinatorics
15A18Eigenvalues, singular values, and eigenvectors
References:
[1]Carlitz, L.: The characteristic polynomial of a certain matrix of binomial coefficients, Fibonacci quart. 3, 81-89 (1965) · Zbl 0125.28204
[2]Chen, W. Y. C.; Louck, J. D.: The combinatorial power of the companion matrix, Linear algebra appl. 232, 261-278 (1996) · Zbl 0838.15015 · doi:10.1016/0024-3795(95)90163-9
[3]Cooper, C.; Kennedy, R.: Proof of a result by jarden by generalizing a proof by carlitz, Fibonacci quart. 33.4, 304-310 (1995) · Zbl 0827.11009
[4]Duvall, P.; Vaughan, T.: Pell polynomials and a conjecture of mahon and horadam, Fibonacci quart. 26, No. 4, 344-353 (1988) · Zbl 0659.10011
[5]Edelman, A.; Strang, G.: Pascal matrices, Amer. math. Monthly 111, No. 3, 189-197 (2004) · Zbl 1089.15025 · doi:10.2307/4145127
[6]Er, M. C.: Sums of Fibonacci numbers by matrix methods, Fibonacci quart. 22, No. 3, 204-207 (1984) · Zbl 0539.10013
[7]Gould, H. W.: The bracket function and fountené–ward generalized binomial coefficients with application to fibonomial coefficients, Fibonacci quart. 7, 23-40 (1969) · Zbl 0191.32702
[8]Hillman, A. P.; Hoggatt, V. E.: The characteristic polynomial of the generalized shift matrix, Fibonacci quart. 3, No. 2, 91-94 (1965) · Zbl 0129.01002
[9]Jr., V. E. Hoggatt: Fibonacci numbers and generalized binomial coefficients, Fibonacci quart. 5, 383-400 (1967) · Zbl 0157.03101
[10]Jr., V. Hoggatt; Bicknell, M.: Fourth power Fibonacci identities from Pascal’s triangle, Fibonacci quart. 2, 81-89 (1964) · Zbl 0125.02102
[11]Horadam, A. F.: Generating functions for powers of a certain generalized sequence of numbers, Duke math. J. 32, 437-446 (1965) · Zbl 0131.04104 · doi:10.1215/S0012-7094-65-03244-8
[12]Jarden, D.: Recurring sequences, (1958)
[13]Jarden, D.; Motzkin, T.: The product of sequences with a common linear recursion formula of order 2, Riveon lematematika 3, 25-27 (1949)
[14]Kalman, D.: Generalized Fibonacci numbers by matrix methods, Fibonacci quart. 20, No. 1, 73-76 (1982) · Zbl 0472.10016
[15]Kilic, E.: Sums of the squares of terms of sequence {un}, Proc. indian acad. Sci. (Math. Sci.) 118, No. 1, 27-41 (2008) · Zbl 1206.11018 · doi:10.1007/s12044-008-0003-y
[16]Koshy, T.: Fibonacci and Lucas numbers with applications, Pure and applied mathematics (2001) · Zbl 0984.11010
[17]Lind, D. A.: A determinant involving generalized binomial coefficients, Fibonacci quart. 9, No. 2, 113-119 (1971) · Zbl 0221.05013
[18]Minc, H.: Permanents of (0,1)-circulants, Canad. math. Bull. 7, No. 2, 253-263 (1964)
[19]Seibert, J.; Trojovsky, P.: On some identities for the fibonomial coefficients, Math. slovaca 55, 9-19 (2005) · Zbl 1108.11019
[20]E. Kilic, P. Stanica, G.N. Stanica, Spectral properties of some combinatorial matrices, in: 13th International Conference on Fibonacci Numbers and Their Applications, 2008
[21]Stanica, P.: Netted matrices, Int. J. Math. math. Sci. 39, 2507-2518 (2003)
[22]Strang, G.: Introduction to linear algebra, (2003)
[23]Torretto, R.: A. Fuchs and generalized binomial coefficients, Fibonacci quart. 2, 296-302 (1964) · Zbl 0129.02502
[24]Trojovsky, P.: Pavel on some identities for the fibonomial coefficients via generating function, Discrete appl. Math. 155, No. 15, 2017-2024 (2007) · Zbl 1144.11018 · doi:10.1016/j.dam.2007.05.003