Özkoç Advances in Difference Equations (2015) 2015:148 DOI 10.1186/s13662-015-0486-7
R E S E A R C H Open Access Some algebraic identities on quadra Fibona-Pell integer sequence Arzu Özkoç*
*Correspondence: [email protected] Abstract Department of Mathematics, Faculty of Arts and Science, Düzce In this work, we define a quadra Fibona-Pell integer sequence University,Düzce,Turkey Wn =3Wn–1 –3Wn–3 – Wn–4 for n ≥ 4 with initial values W0 = W1 =0,W2 =1,W3 =3, and we derive some algebraic identities on it including its relationship with Fibonacci and Pell numbers. Keywords: Fibonacci numbers; Lucas numbers; Pell numbers; Binet’s formula; binary linear recurrences
1Preliminaries Let p and q be non-zero integers such that D = p –q = (to exclude a degenerate case).
We set the sequences Un and Vn to be
Un = Un(p, q)=pUn– – qUn–, () Vn = Vn(p, q)=pVn– – qVn–
for n ≥ with initial values U =,U =,V =,andV = p.ThesequencesUn and Vn
are called the (first and second) Lucas sequences with parameters p and q. Vn is also called the companion Lucas sequence with parameters p and q. The characteristic√ equation√ of Un and Vn is x – px + q = and hence the roots of it are p+ D p– D x = and x = .SotheirBinetformulasare
n n x – x n n Un = and Vn = x + x x – x ≥ p –q for n . For the companion matrix M = ,onehas U V p n = Mn– and n = Mn– Un– Vn–
for n ≥ . The generating functions of Un and Vn are
x –px U(x)= and V(x)= .() –px + qx –px + qx
Fibonacci, Lucas, Pell, and Pell-Lucas numbers can be derived from (). Indeed for p =
and q =–,thenumbersUn = Un(,–) are called the Fibonacci numbers (A in
© 2015 Özkoç; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Özkoç Advances in Difference Equations (2015) 2015:148 Page 2 of 10
OEIS), while the numbers Vn = Vn(,–) are called the Lucas numbers (A in OEIS).
Similarly, for p =andq =–,thenumbersUn = Un(, –) are called the Pell numbers
(A in OEIS), while the numbers Vn = Vn(, –) are called the Pell-Lucas (A in OEIS) (companion Pell) numbers (for further details see [–]).
2 Quadra Fibona-Pell sequence In [], the author considered the quadra Pell numbers D(n), which are the numbers of the form D(n)=D(n –)+D(n –)+D(n –)forn ≥ with initial values D() = D() = D() = , D() = , and the author derived some algebraic relations on it.
In [], the authors considered the integer sequence (with four parameters) Tn =–Tn– –
Tn– +Tn– +Tn– with initial values T =,T =,T =–,T =,andtheyderived some algebraic relations on it. In the present paper, we want to define a similar sequence related to Fibonacci and Pell numbers and derive some algebraic relations on it. For this reason, we set the integer se-
quence Wn to be
Wn =Wn– –Wn– – Wn– ()
for n ≥ with initial values W = W =,W =,W = and call it a quadra Fibona- Pell sequence. Here one may wonder why we choose this equation and call it a quadra Fibona-Pell sequence. Let us explain: We will see below that the roots of the characteristic
equation of Wn are the roots of the characteristic equations of both Fibonacci and Pell sequences. Indeed, the characteristic equation of ()isx –x +x +=andhencethe roots of it are √ √ + – √ √ α = , β = , γ =+ andδ =– . ()
(Here α, β are the roots of the characteristic equation of Fibonacci numbers and γ , δ are the roots of the characteristic equation of Pell numbers.) Then we can give the following
results for Wn.
Theorem The generating function for Wn is
x W(x)= . x +x –x +
Proof The generating function W(x) is a function whose formal power series expansion at x = has the form
∞ n n W(x)= Wnx = W + Wx + Wx + ···+ Wnx + ··· . n=
Since the characteristic equation of ()isx –x +x +=,weget n –x +x + x W(x)= –x +x + x W + Wx + ···+ Wnx + ···
= W +(W –W)x +(W –W)x Özkoç Advances in Difference Equations (2015) 2015:148 Page 3 of 10
+(W –W +W)x + ···
n +(Wn –Wn– +Wn– + Wn–)x + ··· .
Notice that W = W =,W =,W =,andWn =Wn– –Wn– – Wn–.So(–x + x + x)W(x)=x and hence the result is obvious.
Theorem The Binet formula for Wn is γ n – δn αn – βn W = – n γ – δ α – β
for n ≥ .
x Proof Note that the generating function is W(x)= x+x–x+ .Itiseasilyseenthatx + x –x +=(–x – x)( – x – x). So we can rewrite W(x)as
x x W(x)= – .() –x – x –x – x
From (), we see that the generating function for Pell numbers is
x P(x)= () –x – x
and the generating function for the Fibonacci numbers is
x F(x)= .() –x – x
γ n–δn αn–βn From (), (), (), we get W(x)=P(x)–F(x). So Wn =( γ –δ )–( α–β )aswewanted.
The relationship with Fibonacci, Lucas, and Pell numbers is given below.
Theorem For the sequences Wn, Fn, Ln, and Pn, we have:
() Wn = Pn – Fn for n ≥ . n n n n ≥ () W√ n+ + W√n– =(γ + δ )–(α + β ) for n . n n n n () Fn + Pn =(γ – δ )+(α – β ) for n ≥ . n n n n () Ln + Pn+ + Pn– = α + β + γ + δ for n ≥ . n n () (Wn+ – Wn + Fn–)=γ + δ for n ≥ . Wn () limn→∞ = γ . Wn–
Proof () It is clear from the above theorem, since W(x)=P(x)–F(x).
() Since Wn– + Wn+ =Wn+ –Wn– – Wn– +Wn–,weget
W + W =W + W + W n+ n– n– n– n+ γ n– – δn– αn– – βn– = – γ – δ α – β γ n– – δn– αn– – βn– + – γ – δ α – β Özkoç Advances in Difference Equations (2015) 2015:148 Page 4 of 10
γ n+ – δn+ αn+ – βn+ + – γ – δ α – β – = γ n + + γ + δn – – δ (γ – δ) γ γ δ δ – + αn – – α + βn + + β (α – β) α α β β = γ n + δn – αn + βn , √ √ – – since γ + γ + γ = δ – δ – δ = and α – α – α = β + β + β =– . n n n n √ √ () Notice that F = α –β and P = γ –δ .Soweget F = αn –βn and P = γ n –δn. √ n √α–β n γ –δ n n n n n n Thus clearly, Fn + Pn =(γ – δ )+(α – β ). n n n n () It is easily seen that Pn+ + Pn– = γ + δ .AlsoLn = α + β .SoLn + Pn+ + Pn– = αn + βn + γ n + δn.
() Since Wn+ =Wn –Wn– – Wn–,weeasilyget
Wn+ – Wn =Wn –Wn– – Wn– γ n – δn αn – βn γ n– – δn– αn– – βn– = – – – γ – δ α – β γ – δ α – β γ n– – δn– αn– – βn– – – γ – δ α – β = γ n – – + δn – + + γ – δ γ γ δ δ + αn– α – – – βn– β – – α – β α α β β
and hence n γ –γ – n –δ +δ + Wn+ –Wn = √ γ + δ γ δ α –α – β –β – – αn– – βn– α – β α β α –α – β –β – ⇔ W –W + αn– – βn– n+ n α – β α β γ –γ – –δ +δ + = √ γ n + δn γ δ n n ⇔ (Wn+ – Wn + Fn–)=γ + δ , √ γ –γ – –δ+δ+ α–α– β–β– since γ = δ = and α = β =. γ n–δn αn–βn ()Itisjustanalgebraiccomputation,sinceWn =( γ –δ )–( α–β ).
Theorem ThesumofthefirstntermsofWn is
n W +W +W + W + W = n n– n– n– () i i=
for n ≥ . Özkoç Advances in Difference Equations (2015) 2015:148 Page 5 of 10
Proof Recall that Wn =Wn– –Wn– – Wn–.So
Wn– + Wn– =Wn– –Wn– – Wn.()
Applying (), we deduce that
W + W =W –W – W,
W + W =W –W – W,
W + W =W –W – W, () ...,
Wn– + Wn– =Wn– –Wn– – Wn–,
Wn– + Wn– =Wn– –Wn– – Wn.
Ifwesumofbothsidesof(), then we obtain Wn– + W +(W + ···+ Wn–)=(W +
W + ···+ Wn–)–(W + W + ···+ Wn–)–(W + W + ···+ Wn). So we get Wn– +(W +
W + ···+ Wn–)=–Wn– – Wn– – Wn +Wn– +Wn– and hence we get the desired result.
Theorem The recurrence relations are
Wn =Wn– –Wn– +Wn– – Wn–,
Wn+ =Wn– –Wn– +Wn– – Wn–
for n ≥ .
Proof Recall that Wn =Wn– –Wn– – Wn–.SoWn =Wn– –Wn– – Wn– and hence
Wn =Wn– –Wn– – Wn–
=Wn– –Wn– –Wn– –Wn– +Wn– +Wn–
+ Wn– – Wn– – Wn–
=–(Wn– –Wn– – Wn–)+Wn– –Wn– +Wn–
– Wn– – Wn–
=–Wn– +Wn– –Wn– –Wn– +Wn– – Wn– – Wn–
=Wn– –Wn– +Wn– – Wn–.
The other assertion can be proved similarly.
The rank of an integer N is defined to be p if p isthesmallestprimewithp|N, ρ(N)= ∞ if N is prime.
Thus we can give the following theorem. Özkoç Advances in Difference Equations (2015) 2015:148 Page 6 of 10
Theorem The rank of Wn is ⎧ ⎨⎪ if n =+k,+k,+k, ρ(W )= if n =+k,+k,+k,+k, n ⎩⎪ if n =+k,+k
for an integer k ≥ .
Proof Let n =+k. We prove it by induction on k.Letk =.ThenwegetW == ·.
So ρ(W) = . Let us assume that the rank of Wn is for n = k –,thatis,ρ(Wk–)=,so a W+(k–) = Wk– = · B for some integers a ≥ andB >.Forn = k,weget
Wk+ =Wk+ –Wk+ – Wk+
=(Wk+ –Wk+ – Wk)–Wk+ – Wk+
=Wk+ –Wk+ –Wk –Wk+ – Wk+
=(Wk+ –Wk – Wk–)–Wk+ –Wk –Wk+ – Wk+
=Wk+ –Wk –Wk– –Wk+ –Wk –Wk+ – Wk+
=Wk+ –Wk –Wk+ –Wk–
a =Wk+ –Wk –Wk+ –· B a– = Wk+ –Wk –Wk+ –· B .
Therefore ρ(W+k) = . Similarly it can be shown that ρ(W+k)=ρ(W+k)=. Now let n =+k.Fork =,wegetW == · . So ρ(W)=.Letusassume b that for n = k –therankofWn is , that is, ρ(W+(k–))=ρ(Wk–)= · H for some integers b ≥ andH > which is not even integer. For n = k,weget
Wk+ =Wk+ –Wk+ – Wk+
=Wk+ –Wk+ –(Wk+ –Wk+ – Wk)
=Wk+ –Wk+ –Wk+ +Wk+ + Wk
=Wk+ –Wk+ –Wk+ +Wk+
+(Wk– –Wk– – Wk–)
=Wk+ –Wk+ –Wk+ +Wk+ +Wk–
–Wk– – Wk–
=Wk+ –Wk+ –Wk+ +Wk+ +Wk–
b –Wk– – · H = Wk+ – Wk+ – Wk+ + Wk+ + Wk– b– – Wk– – · H .
So ρ(Wk+) = . The others can be proved similarly. Özkoç Advances in Difference Equations (2015) 2015:148 Page 7 of 10
Remark Apart from the above theorem, we see that ρ(W)=ρ(W)=∞,while
ρ(W)=ρ(W) = and ρ(W)=ρ(W)=ρ(W) = . But there is no general for- mula.
The companion matrix for Wn is ⎡ ⎤ – – ⎢ ⎥ ⎢ ⎥ M = ⎢ ⎥ . ⎣ ⎦
Set ⎡ ⎤ ⎢ ⎥ ⎢⎥ N = ⎢ ⎥ ⎣⎦
and
R =[].
Then we can give the following theorem, which can be proved by induction on n.
Theorem For the sequence Wn, we have: n () RM N = Wn+ + Pn +(Wn+ – Fn) for n ≥ . T n– () R(M ) N = Wn for n ≥ . () If n ≥ is odd, then ⎡ ⎤ m m m m ⎢ ⎥ ⎢ ⎥ n ⎢m m m m⎥ M = ⎣ ⎦ , m m m m
m m m m
where
m = Wn+, m = Wn+, m = Wn, m = Wn–,
m =–Wn+, m =–Wn, m =–Wn–, m =–Wn–,
n– n– m =––Wn+ – Wn––i, m =–Wn+ – Wn–i, i= i=
n– n– m =–Wn – Wn––i, m =––Wn+ – Wn––i, i= i=
n– n– m =––Wn– – Wn––i, m =–Wn – Wn––i, i= i= Özkoç Advances in Difference Equations (2015) 2015:148 Page 8 of 10
n– n– m =–Wn– – Wn––i, m =––Wn– – Wn––i, i= i=
and if n ≥ is even, then
⎡ ⎤ m m m m ⎢ ⎥ ⎢ ⎥ n ⎢m m m m⎥ M = ⎣ ⎦ , m m m m
m m m m
where
m = Wn+, m = Wn+, m = Wn, m = Wn–,
m =–Wn+, m =–Wn, m =–Wn–, m =–Wn–,
n– n– m =–Wn+ – Wn––i, m =––Wn+ – Wn–i, i= i=
n– n– m =––Wn – Wn––i, m =–Wn+ – Wn––i, i= i=
n– n– m =–Wn– – Wn––i, m =––Wn – Wn––i, i= i=
n– n– m =––Wn– – Wn––i, m =–Wn– – Wn––i. i= i=
AcirculantmatrixisamatrixA =[aij]n×n defined to be
⎡ ⎤ a a a ··· a ⎢ n–⎥ ⎢ ··· ⎥ ⎢an– a a an–⎥ ⎢ ··· ⎥ ⎢an– an– a an–⎥ A = ⎢ ⎥ , ⎢ · · · ··· · ⎥ ⎢ ⎥ ⎣ · · · ··· · ⎦
a a a ··· a
where ai are constants. The eigenvalues of A are
n– –jk λj(A)= akw ,() k=
√ πi where w = e n , i = –, and j =,,...,n – . The spectral norm for a matrix B =[b ] × √ ij n m H is defined to be Bspec = max{ λi},whereλi are the eigenvalues of B B for ≤ j ≤ n – and BH denotes the conjugate transpose of B. Özkoç Advances in Difference Equations (2015) 2015:148 Page 9 of 10
For the circulant matrix ⎡ ⎤ W W W ··· W ⎢ n–⎥ ⎢ ··· ⎥ ⎢Wn– W W Wn–⎥ ⎢ ··· ⎥ ⎢Wn– Wn– W Wn–⎥ W = W(Wn)=⎢ ⎥ ⎢ · · · ··· · ⎥ ⎢ ⎥ ⎣ · · · ··· · ⎦
W W W ··· W
for Wn, we can give the following theorem.
Theorem The eigenvalues of W are –j –j Wn–w +(Wn + Pn– –Fn– +)w –j +(Pn –Fn – Wn–)w – Wn λ (W)= j w–j +w–j –w–j +
for j =,,,...,n –.
Proof Applying ()weeasilyget
n– n– γ k – δk αk – βk λ (W)= W w–jk = – w–jk j k γ – δ α – β k= k= γ n – δn – αn – βn – = – – – γ – δ γ w–j – δw–j – α – β αw–j – βw–j – (γ n –)(δw–j –)–(δn –)(γ w–j –) = γ – δ (γ w–j –)(δw–j –) (αn –)(βw–j –)–(βn –)(αw–j –) – α – β (αw–j –)(βw–j –) w–j(γ nδ – δnγ + γ – δ)+δn – γ n = γ – δ δγw–j – w–j(δ + γ )+ w–j(αnβ – βnα + α – β)+βn – αn – α – β βαw–j – w–j(β + α)+ ⎧ √ √ ⎫ ⎪w–j[ (δ – γ + γδn – δγ n)+ (α – β + αnβ – αβ n)] ⎪ ⎪ √ √ ⎪ ⎪ –j n n n n n n ⎪ ⎨ + w√ [ (γ – δ + δ – γ + γδ –√γ δ)+ (β – α ) ⎬ n n –j n n ⎪ + (α – β + α√β – αβ )] + w [ (γ – δ√+ γ – δ ⎪ ⎪ ⎪ ⎪ + γ nδ – γδn)+ (β – α + βnα – αnβ)+ (βn – αn)]⎪ ⎩⎪ √ √ ⎭⎪ +[ (δn – γ n)+ (αn – βn)] = √ (w–j +w–j –w–j +) –j –j Wn–w +(Wn + Pn– –Fn– +)w –j +(Pn –Fn – Wn–)w – Wn = , w–j +w–j –w–j + √ √ since αβ =–,γδ=–,α + β =,α – β = , γ + δ =,andγ – δ = . Özkoç Advances in Difference Equations (2015) 2015:148 Page 10 of 10
After all, we consider the spectral norm of W.Letn =.ThenW =[]×.So
Wspec = . Similarly for n =,weget ⎡ ⎤ ⎢ ⎥ W = ⎣⎦
H ≥ and hence W W = I.So W spec =.Forn , the spectral norm of Wn is given by the following theorem, which can be proved by induction on n.
Theorem The spectral norm of Wn is
W +W +W + W + W = n– n– n– n– n spec
for n ≥ .
H For example, let n = . Then the eigenvalues of W W are
λ =,, λ =, λ = λ = and λ = λ =. √ W+W+W+W+ So the spectral norm is W spec = λ =.Also =.Consequently,
W +W +W + W + W = = spec
as we claimed.
Competing interests The author declares that they have no competing interests.
Acknowledgements The author wishes to thank Professor Ahmet Tekcan of Uludag University for constructive suggestions.
Received: 25 March 2015 Accepted: 26 April 2015
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