A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence

Abstract

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.

1. Introduction

The generalized Fibonacci sequence {𝑈_𝑛} and the generalized Lucas sequence {𝑉_𝑛} are defined for 𝑛>1, by, 𝑈_𝑛=𝑝𝑈_𝑛−1+𝑈_𝑛−2,𝑉_𝑛=𝑝𝑉_𝑛−1+𝑉_𝑛−2, (1) where 𝑈₀=0, 𝑈₁=1 and 𝑉₀=2,𝑉₁=𝑝, respectively.

Let 𝛼 and 𝛽 be the roots of the equation 𝑥²−𝑝𝑥−1=0. Then the Binet formulas of the sequences {𝑈_𝑛} and {𝑉_𝑛} are given by 𝑈_𝑛=𝛼^𝑛−𝛽^𝑛𝛼−𝛽,𝑉_𝑛=𝛼^𝑛+𝛽^𝑛. (2) If 𝑝=1, then 𝑈_𝑛=𝐹_𝑛 (𝑛th Fibonacci number) and 𝑉_𝑛=𝐿_𝑛 (𝑛th Lucas number).

It is a well-known fact that gcd𝐹_𝑛,𝐹_𝑚=𝐹_{gcd(𝑛,𝑚)}. (3) It is also known that 𝐹_𝑘𝑛 is a multiple of 𝐹_𝑛, for all integers 𝑘 and 𝑛. In [1], the author showed that, for 𝑛>2, the Fibonacci number 𝐹_𝑚 is a multiplication of 𝐹²_𝑛 if and only if 𝑚 is multiplication of 𝑛𝐹_𝑛 (for more details see [2]). Also, in [3], the author obtained the following divisibility properties:(i)𝐹_𝑘𝑛−1−𝐹^𝑘_𝑛−1 is divisible by 𝐹²_𝑛;(ii)𝐹_𝑘𝑛−2−(−1)^𝑘+1𝐹^𝑘_𝑛−2 is divisible by 𝐹²_𝑛,where 𝑛,𝑘≥1. Kiliç [4] generalized these results for a general second-order linear recursion {𝑈_𝑛} as follows: 𝑈_𝑟^𝑘−1𝑈_{𝑘𝑛−𝑟}−(−1)^{(𝑟−1)(𝑘+1)}𝑈^𝑘_𝑛−𝑟isdivisibleby𝑈²_𝑛. (4)

In this paper, we investigate divisibility properties of the generalized Fibonacci numbers by 𝑈^𝑙_𝑛, where 𝑙≥3. For 𝑙=3, we show that 𝑈₂^𝑘−2𝑈_𝑘𝑛−3−𝑈₂^𝑘−2𝑈^𝑘_𝑛−1+(−1)^𝑘−1𝑈^𝑘_𝑛−2isdivisibleby𝑈³_𝑛. (5) We use matrix methods to prove the claim. We recall that matrix methods are useful tools for deriving some properties of linear recurrences (see [4–9]). We consider the quotient 𝑈₂^𝑘−2𝑈_𝑘𝑛−3−𝑈₂^𝑘−2𝑈^𝑘_𝑛−1+(−1)^𝑘−1𝑈^𝑘_𝑛−2𝑈³_𝑛 (6) for all positive integers 𝑛 and 𝑘. We define a generating matrix for this quotient for fixed 𝑛 and increasing values of 𝑘. 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 𝑙>3. Thus we obtain a generalization of the results given in [4].

2. Main Results

We denote the quotient (𝑈₂^𝑘−2𝑈_𝑘𝑛−3−𝑈₂^𝑘−2𝑈^𝑘_𝑛−1+(−1)^𝑘−1𝑈^𝑘_𝑛−2)/𝑈³_𝑛 by 𝑠(𝑛,𝑘).

Define a second-order linear sequence {𝐴_𝑛}, for 𝑛>1, 𝐴_𝑛=𝑝𝐴_𝑛−1+𝐴_𝑛−2 (7) with initial conditions 𝐴₀=𝑝²−1 and 𝐴₁=𝑝³+3𝑝.

By the definitions of {𝑈_𝑛} and {𝐴_𝑛}, we have 𝐴_𝑛=𝑈₂𝑉_𝑛+1+𝑈₂𝑈_𝑛−𝑈_𝑛−1. (8)

Define a matrix 𝐻(𝑛) by 𝐻(𝑛)=𝐴_𝑛−1𝐵_𝑛𝐶_𝑛(−1)^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2100001000010, (9) where 𝐵_𝑛=𝑈₂𝑈_𝑛−2−𝑈²₂𝑈_𝑛−1𝑉_𝑛+𝑈²_𝑛−1−𝑈²_𝑛−2+(−1)^𝑛+1𝑈₃,𝐶_𝑛=−𝑈²₂𝑈_𝑛−1𝑈_𝑛−2𝑉_𝑛+(−1)^𝑛+1𝑈²₂𝑈_𝑛−2+(−1)^𝑛𝑈³₂𝑈_𝑛−1. (10)

We next define a matrix 𝐺(𝑛,𝑘) of order 4 as follows: 𝐺()=𝑠()𝑡()𝑦()𝑛,𝑘𝑛,𝑘+3𝑛,𝑘+3𝑛,𝑘+3)(−1^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2𝑠()𝑠()𝑡()𝑦()𝑛,𝑘+2𝑛,𝑘+2𝑛,𝑘+2𝑛,𝑘+2)(−1^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2𝑠()𝑠()𝑡()𝑦()𝑛,𝑘+1𝑛,𝑘+1𝑛,𝑘+1𝑛,𝑘+1)(−1^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2𝑠()𝑠()𝑡()𝑦()𝑛,𝑘𝑛,𝑘𝑛,𝑘𝑛,𝑘)(−1^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2𝑠(𝑛,𝑘−1). (11)

𝑡(𝑛,𝑘) and 𝑦(𝑛,𝑘) are given by 𝑡(𝑛,𝑘)=−𝑈₂^𝑘−2𝑈_𝑛−1𝑈_𝑛−2𝛿_𝑛𝑈_𝑘𝑛+(−1)^𝑛+1𝑈₂^𝑘−2𝑈₂𝑈_𝑛−1𝛿_𝑛+𝑈²_𝑛−2𝑈_{𝑛(𝑘−1)}+(−1)^3𝑛𝑈^𝑘₂𝑈_{𝑛(𝑘−3)}+𝑈₂^𝑘−2𝑈²_𝑛2(−1)^𝑛𝑈_−𝑛+2+𝑈_𝑛+2𝑈^𝑘_𝑛−1+(−1)^𝑘𝑈₂𝑈²_𝑛2(−1)^𝑛𝑈_−𝑛+1+𝑈_𝑛+1𝑈^𝑘_𝑛−2×𝑈⁵_𝑛⁻¹,𝑦(𝑛,𝑘)=−𝑈₂^𝑘−1𝑈²_𝑛−1𝑈²_𝑛−2𝑈_𝑘𝑛+𝑈₂^𝑘−1𝑈₂𝑈_𝑛−1𝛿_𝑛+𝑈²_𝑛−2𝑈_{𝑛(𝑘−2)}+(−1)^3𝑛+1𝑈^𝑘₂𝛿_𝑛𝑈_{𝑛(𝑘−3)}+𝑈²_𝑛2(−1)^𝑛+1𝑈₂+𝑈_2𝑛−2𝑈^𝑘_𝑛−1+(−1)^𝑘𝑈²₂𝑈²_𝑛2(−1)^𝑛+1−𝑈_2𝑛−1𝑈^𝑘_𝑛−2×𝑈⁵_𝑛⁻¹, (12) where 𝛿_𝑛=𝑈₂𝑈_𝑛−1−𝑈_𝑛−2. (13) Thus we give our first main result.

Theorem 1. For 𝑛≥1, 𝐻(𝑛)^𝑘=𝐺(𝑛,𝑘). (14)

Proof. We will use induction on 𝑘. The result is clear for 𝑘=1. Now assume that 𝐻(𝑛)^𝑘−1=𝐺(𝑛,𝑘−1). Then, by the definitions of 𝑠(𝑛,𝑘),𝑡(𝑛,𝑘), and 𝑦(𝑛,𝑘), we have 𝐴_𝑛−1𝑠()()(),𝐵𝑛,𝑘+2+𝑡𝑛,𝑘+2=𝑠𝑛,𝑘+3_𝑛𝑠()()(),𝐶𝑛,𝑘+2+𝑦𝑛,𝑘+2=𝑡𝑛,𝑘+3_𝑛𝑠(𝑛,𝑘+2)+(−1)^𝑛𝑈³₂𝑈_𝑛−1𝑈_𝑛−2𝑠()().𝑛,𝑘+1=𝑦𝑛,𝑘+3 (15) Thus the proof is complete.

As a consequence of this theorem, we can see that matrix 𝐻(𝑛) generates 𝑠(𝑛,𝑘). Since the elements of 𝐻(𝑛) are integers, the quotient 𝑠(𝑛,𝑘) are integers for all positive integers 𝑛 and 𝑘.

Lemma 2. For 𝑛≥1, the eigenvalues of 𝐻(𝑛) are 𝑈₂𝛼^𝑛, 𝑈₂𝛽^𝑛, 𝑈₂𝑈_𝑛−1, and −𝑈_𝑛−2.

Proof. The characteristic polynomial of 𝐻(𝑛) is 𝑥⁴−𝐴_𝑛−1𝑥³−𝐵_𝑛𝑥²−𝐶_𝑛𝑥+(−1)^𝑛+1𝑈³₂𝑈_𝑛−1𝑈_𝑛−2=0, (16) and it is factorized as 𝑥−𝑈₂𝛼^𝑛𝑥−𝑈₂𝛽^𝑛𝑥−𝑈₂𝑈_𝑛−1𝑥+𝑈_𝑛−2=0, (17) which completes the proof.

As another main result, we have the following theorem.

Theorem 3. For 𝑛,𝑘≥1, (𝐺(𝑛,𝑘))_4,1()=𝑈=𝑠𝑛,𝑘_2𝑘−2𝑈_𝑛−1𝑈_𝑛−2𝑈_𝑘𝑛+(−1)^𝑛𝑈₂^𝑘−2𝛿_𝑛𝑈_{𝑛(𝑘−1)}−𝑈₂^𝑘−1𝑈_{𝑛(𝑘−2)}−𝑈₂^𝑘−2𝑈²_𝑛𝑈^𝑘_𝑛−1+(−1)^𝑘−1𝑈²_𝑛𝑈^𝑘_𝑛−2×𝑈⁵_𝑛⁻¹, (18) where 𝛿_𝑛 is defined as shown previously.

Proof. Since the eigenvalues of 𝐻(𝑛) are distinct, 𝐻(𝑛) is diagonalizable as 𝑉⁻¹𝐻(𝑛)𝑉=𝐷, (19) where 𝑈𝑉=³²𝛼^3𝑛𝑈³₂𝛽^3𝑛𝑈³₂𝑈³_𝑛−1−𝑈³_𝑛−2𝑈²₂𝛼^2𝑛𝑈²₂𝛽^2𝑛𝑈²₂𝑈²_𝑛−1𝑈²_𝑛−2𝑈₂𝛼^𝑛𝑈₂𝛽^𝑛𝑈₂𝑈_𝑛−1−𝑈_𝑛−21111 (20) and 𝐷=diag(𝑈₂𝛼^𝑛,𝑈₂𝛽^𝑛,𝑈₂𝑈_𝑛−1,−𝑈_𝑛−2). Therefore, we obtain 𝑉⁻¹𝐻(𝑛)^𝑘𝑉=𝐷^𝑘. By Theorem 1, we write 𝑉⁻¹𝐺(𝑛,𝑘)𝑉=𝐷^𝑘. Then we have the following linear equation system: 𝑔_𝑖1𝑈³₂𝛼^3𝑛+𝑔_𝑖2𝑈²₂𝛼^2𝑛+𝑔_𝑖3𝑈₂𝛼^𝑛+𝑔_𝑖4=𝑈₂^{𝑘−𝑖+4}𝛼^{(𝑘−𝑖+4)𝑛},𝑔_𝑖1𝑈³₂𝛽^3𝑛+𝑔_𝑖2𝑈²₂𝛽^2𝑛+𝑔_𝑖3𝑈₂𝛽^𝑛+𝑔_𝑖4=𝑈₂^{𝑘−𝑖+4}𝛽^{(𝑘−𝑖+4)𝑛},𝑔_𝑖1𝑈³₂𝑈³_𝑛−1+𝑔_𝑖2𝑈²₂𝑈²_𝑛−1+𝑔_𝑖3𝑈₂𝑈_𝑛−1+𝑔_𝑖4=𝑈₂^{𝑘−𝑖+4}𝑈^{𝑘−𝑖+4}_𝑛−1,−𝑔_𝑖1𝑈³_𝑛−2+𝑔_𝑖2𝑈²_𝑛−2−𝑔_𝑖3𝑈_𝑛−2+𝑔_𝑖4=(−1)^{𝑘−𝑖+4}𝑈^{𝑘−𝑖+4}_𝑛−2. (21) The solution of the above linear equation system gives the claimed result.

By considering the definition of 𝑠(𝑛,𝑘), we have the following consequence of Theorem 3.

Corollary 4. For 𝑛,𝑘≥1, 𝑈_𝑛−1𝑈_𝑛−2𝑈_𝑘𝑛=𝑈²_𝑛𝑈_𝑘𝑛−3+(−1)^𝑛+1𝛿_𝑛𝑈_{𝑛(𝑘−1)}+𝑈₂𝑈_{𝑛(𝑘−2)}. (22)

The next results generalize the result given by Corollary 4.

Theorem 5. For all integers 𝑟, 𝑈_𝑛−𝑟𝑈_{𝑛−𝑟−1}𝑈_𝑘𝑛=𝑈²_𝑛𝑈_{𝑘𝑛−2𝑟−1}+(−1)^𝑛+𝑟𝑈_𝑟+1𝑈_𝑛−𝑟−𝑈_𝑟𝑈_{𝑛−𝑟−1}𝑈_{𝑛(𝑘−1)}+𝑈_𝑟𝑈_𝑟+1𝑈_{𝑛(𝑘−2)},𝑈_𝑛−𝑟𝑈_{𝑛−𝑟−1}𝑉_𝑘𝑛=𝑈²_𝑛𝑉_{𝑘𝑛−2𝑟−1}+(−1)^𝑛+𝑟𝑈_𝑟+1𝑈_𝑛−𝑟−𝑈_𝑟𝑈_{𝑛−𝑟−1}𝑉_{𝑛(𝑘−1)}+𝑈_𝑟𝑈_𝑟+1𝑉_{𝑛(𝑘−2)}. (23)

Proof. The proof can be seen by the Binet formulas of the sequences {𝑈_𝑛} and {𝑉_𝑛}.

For 𝑙=3, we give the general case of divisibility properties in the following result.

Corollary 6. For all integers 𝑟, 𝑈_𝑟^𝑘−2𝑈^𝑘−2_𝑟+1𝑈_{𝑘𝑛−2𝑟−1}−(−1)^{𝑘(𝑟−1)}𝑈^𝑘−2_𝑟+1𝑈^𝑘_𝑛−𝑟+(−1)^𝑘𝑟−1𝑈_𝑟^𝑘−2𝑈^𝑘_{𝑛−𝑟−1} (24) is divisible by 𝑈³_𝑛.

3. Generalization of the Divisibility Properties

In this section, for a positive integer 𝑙, we generalize divisibility properties. For this purpose we introduce some new notations.

Let 𝑟_𝑖 be an integer for 𝑖=1,…,𝑙−1. Let 𝜉₁=_1≤∑_𝑖≤𝑚𝑟_𝑖(),∑,𝑚≡1mod2_{1≤𝑖≤𝑚−1}𝑟_𝑖(),𝜉,𝑚≡0mod2₂=_1≤𝑖≤∑_𝑚−1𝑟_𝑖(),∑,𝑚≡1mod2_{1≤𝑖≤𝑚}𝑟_𝑖,𝑚≡0(mod2),𝜑=_{1≤𝑖≤𝑙−1}𝑟_𝑖,𝛾=_{1≤𝑗<𝑖≤𝑙−1}𝑈_{𝑟𝑖−𝑟𝑗}. (25) We denote the above product by 𝛾_𝑚 for 𝑖,𝑗≠𝑚.

Corollary 7. (a) For an even positive integer 𝑙, _{1≤𝑖≤𝑙−1}𝑈_𝑟𝑖^{𝑘−𝑙+1}𝛾𝑈_{𝑘𝑛−𝜑}+_{1≤𝑚≤𝑙−1}(−1)^{𝑘(𝑟𝑚−1)+𝑚+𝜉1+1}𝛾_𝑚𝑈^𝑘_{𝑛−𝑟𝑚1≤𝑖≤𝑙−1𝑚≠𝑖}𝑈_𝑟𝑖^{𝑘−𝑙+1} (26) is divisible by 𝛾𝑈^𝑙_𝑛.
(b) For an odd positive integer 𝑙, _{1≤𝑖≤𝑙−1}𝑈_𝑟𝑖^{𝑘−𝑙+1}𝛾𝑈_{𝑘𝑛−𝜑}+_{1≤𝑚≤𝑙−1}(−1)^{𝑘(𝑟𝑚−1)+𝑚+𝜉2}𝛾_𝑚𝑈^𝑘_{𝑛−𝑟𝑚1≤𝑖≤𝑙−1𝑚≠𝑖}𝑈_𝑟𝑖^{𝑘−𝑙+1} (27) is divisible by 𝛾𝑈^𝑙_𝑛.

As an example, if we take 𝑙=4, 𝑟₁=1, 𝑟₂=2, 𝑟₃=3, and 𝑝=1, then 𝐹₃^𝑘−3𝐹_𝑘𝑛−6−𝐹₃^𝑘−3𝐹^𝑘_𝑛−1+(−1)^𝑘𝐹₃^𝑘−3𝐹^𝑘_𝑛−2+𝐹^𝑘_𝑛−3isdivisibleby𝐹⁴_𝑛. (28)

Acknowledgments

This work is supported by Tubitak and the Scientific Research Projects Office (BAP) of Selcuk University.