A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence

We give divisibility properties of the generalized Fibonacci sequence by matrix methods. We also present new recursive identities for the generalized Fibonacci and Lucas sequences.

It is also known that   is a multiple of   , for all integers  and .In [1], the author showed that, for  > 2, the Fibonacci number   is a multiplication of  2  if and only if  is multiplication of   (for more details see [2]).Also, in [3], the author obtained the following divisibility properties: where ,  ≥ 1. Kilic ¸ [4] generalized these results for a general second-order linear recursion {  } as follows: In this paper, we investigate divisibility properties of the generalized Fibonacci numbers by    , where  ≥ 3.For  = 3, we show that We use matrix methods to prove the claim.We recall that matrix methods are useful tools for deriving some properties of linear recurrences (see [4][5][6][7][8][9]).We consider the quotient for all positive integers  and .We define a generating matrix for this quotient for fixed  and increasing values of .Then we give an explicit statement for the quotient.Also, by considering this explicit statement, we find new recursive identities for the general second-order linear recurrences.Finally, we give divisibility properties of the generalized Fibonacci numbers in the case  > 3. Thus we obtain a generalization of the results given in [4].
By the definitions of {  } and {  }, we have Define a matrix () by where We next define a matrix (, ) of order 4 as follows: (,  + 1)  (,  + 1)  (,  + 1) (−1 (, ) and (, ) are given by where Thus we give our first main result.As a consequence of this theorem, we can see that matrix () generates (, ).Since the elements of () are integers, the quotient (, ) are integers for all positive integers  and .

Lemma 2. For
and it is factorized as which completes the proof.
As another main result, we have the following theorem.
Theorem 3.For ,  ≥ 1, where   is defined as shown previously.
Proof.Since the eigenvalues of () are distinct, () is diagonalizable as where and . Therefore, we obtain  −1 ()   =   .By Theorem 1, we write  −1 (, ) =   .Then we have the following linear equation system: The solution of the above linear equation system gives the claimed result.
By considering the definition of (, ), we have the following consequence of Theorem 3.
The next results generalize the result given by Corollary 4.
Proof.The proof can be seen by the Binet formulas of the sequences {  } and {  }.
For  = 3, we give the general case of divisibility properties in the following result.is divisible by  3  .

Generalization of the Divisibility Properties
In this section, for a positive integer , we generalize divisibility properties.For this purpose we introduce some new notations.