Some identities deriving from the nth power of a special matrix
© Akyuz and Halici; licensee Springer 2012
Received: 12 September 2012
Accepted: 29 November 2012
Published: 21 December 2012
In this paper, we consider the Horadam sequence and some summation formulas involving the terms of the Horadam sequence. We derive combinatorial identities by using the trace, the determinant and the n th power of a special matrix.
1 Introduction and preliminaries
and obtained relations between the Fibonacci Q-matrix and the Lucas matrix.
Then, we derive a formula giving the n th power of this matrix so that its entries involve the generalized Fibonacci and Lucas numbers. We give some identities using this matrix.
2 Some combinatorial identities involving the terms of Horadam sequence
In this section, we derive various combinatorial identities using the equations in (7) and the matrix A. By means of the n th power of the matrix A, we obtain the Cassini-like formula for the generalized Fibonacci and Lucas numbers and give a few different identities.
Hence, we can write . Similarly, the equations provided by the elements and of this matrix can be easily written. When k is even, the proof can be easily seen. Thus, the proof is completed. □
It is noted that Theorem 2.1 generalizes the work in the reference . If we write 1 and −1 instead of p and q in the matrix A, then the matrix A reduces to the Lucas matrix in . Therefore, we can give the following corollary.
where and are the Pell and Pell-Lucas numbers, respectively. Therefore, we obtain some identities related to the Pell and Pell-Lucas numbers. Similarly, we can get some identities related to Jacobsthal and Jacobsthal-Lucas numbers. □
By the aid of the n th power of the matrix A, we can give the relationship between the matrix A and the matrix R as in the following corollary.
Note that these identities are given by using the matrix Q in .
we can write . When , , if the necessary arrangements are made, then the proof is completed. □
Now, using Theorem 2.1, we can also give the following corollary without proof.
Corollary 3 (Cassini-like formula)
respectively, where is as in equation (7).
The authors are very grateful to an anonymous referee for helpful comments and suggestions to improve the presentation of this paper.
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