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On some identities for the Fibonomial coefficients. (English) Zbl 1108.11019
Summary: The Fibonomial coefficients \(\Bigl [\begin{smallmatrix} n\\k\end{smallmatrix} \Bigr ]\) are defined for positive integers \(n\geq k\) as follows \[ \begin{bmatrix} n \\ k \end{bmatrix} =\frac {F_n F_{n-1}\cdots F_{n-k+1}}{F_1 F_2 \cdots F_k}\;, \] with \(\Bigl [\begin{smallmatrix} n\\0 \end{smallmatrix} \Bigr ]=1\), where the Fibonacci numbers are given by the recurrence relation \(F_{n+2}=F_{n+1}+F_{n}\), \(F_0=0\), \(F_1=1\). In this paper new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the \(k\)th powers of the Fibonacci numbers. Their proofs are based on a reasonable manipulation with these generating functions.

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
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[1] CARLITZ L.: Generating functions for powers of a certain sequence of numbers. Duke Math. J. 29 (1962), 521-537. · Zbl 0147.02105
[2] FONTENÉ G.: Généralisation d’une formule connue. Nouv. Ann. (4) 15 (1915), 112.
[3] GOULD H. W.: The bracket function and Fontené Ward generalized binomial coefficients with application to Fibinomial coefficients. Fibonacci Quart. 7 (1969), 23-40. · Zbl 0191.32702
[4] HORADAM A. F.: Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J. 32 (1965), 437-446. · Zbl 0131.04104
[5] HOLTE J. M.: Residues of generalized binomial coefficients modulo primes. Fibonacci Quart. 38 (2000), 227-238. · Zbl 0947.11010
[6] JARDEN D.: Recurring sequences. Riveon Lematimatika (Israel) (1958). · Zbl 0035.30802
[7] RIORDAN J.: Generating functions for powers of Fibonacci numbers. Duke Math. J. 29 (1962), 5-12. · Zbl 0101.28801
[8] SHANNON A. G.: A method of Carlitz applied to the k-th power generating function for Fibonacci numbers. Fibonacci Quart. 12 (1974), 293-299. · Zbl 0287.10010
[9] TORRETTO R.- FUCHS A.: Generalized binomial coefficients. Fibonacci Quart. 2 (1964), 296-302. · Zbl 0129.02502
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