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On the (s,t)-Pell and (s,t)-Pell-Lucas numbers by matrix methods

Somnuk Srisawat, Wanna Sriprad

Department of Mathematics and computer science, Faculty of Science and Technology,

Rajamangala University of Technology Thanyaburi, Pathum Thani 12110,Thailand.

somnuk_s@rmutt.ac.th wanna_sriprad@rmutt.ac.th

Submitted March 17, 2016 — Accepted September 5, 2016

Abstract

In this paper, we investigate some generalization of Pell and Pell-Lucas numbers, which is called (s, t)-Pell and (s, t)-Pell-Lucas numbers, and we define the2×2matrixW,which satisfy the relationW2 = 2sW+tI. After that, we establish some identities of(s, t)-Pell and(s, t)-Pell-Lucas numbers and some sum formulas for(s, t)-Pell and(s, t)-Pell-Lucas numbers by using this matrix.

Keywords:Fibonacci number; Lucas number; Pell number; Pell-Lucas num- ber;(s, t)-Pell number;(s, t)-Pell-Lucas number.

MSC:11B37; 15A15.

1. Introduction

For over several years, there are many recursive sequences that have been studied in the literatures. The famous examples of these sequences are Fibonacci, Lucas, Pell and Pell-Lucas, because they are extensively used in various research areas such as Engineering, Architecture, Nature and Art (for examples see: [2, 3, 4, 5, 6, 7]). For n≥2, the classical Fibonacci{Fn},Lucas{Ln},Pell {Pn} and Pell-Lucas {Qn}

This research was supported by faculty of science and technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thailand.

46(2016) pp. 195–204

http://ami.ektf.hu

195

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sequences are defined byFn=Fn1+Fn2,Ln =Ln1+Ln2,Pn= 2Pn1+Pn2, andQn= 2Qn1+Qn2, with the initial conditionsF0= 0, F1= 1,L0= 2, L1= 1,P0 = 0, P1= 1 and Q0=Q1 = 2, respectively. For more detialed information about Fibonacci, Lucas, Pell, Pell-Lucas sequences can be found in [2, 3].

Recently, Fibonacci, Lucas, Pell and Pell-Lucas were generalized and studied by many authors in the different ways to derive many identities. In 2012, Gulec and Taskara [1] introduced a new generalization of Pell and Pell-Lucas sequence which is called(s, t)-Pell and(s, t)-Pell-Lucas sequences as in the definition 1.1 and by considering these sequences, they introduced the matrix sequences which have elements of(s, t)-Pell and(s, t)-Pell-Lucas sequences. Further, they obtained some properties of(s, t)-Pell and(s, t)-Pell-Lucas matrices sequences by using elementary matrix algebra.

Definition 1.1. [1] Lets, tbe any real number withs2+t >0, s >0 andt6= 0.

Then the (s, t)-Pell sequences {Pn(s, t)}n∈N and the (s, t)-Pell-Lucas sequences {Qn(s, t)}n∈Nare defined respectively by

Pn(s, t) = 2sPn−1(s, t) +tPn−2(s, t), forn≥2, (1.1) Qn(s, t) = 2sQn1(s, t) +tQn2(s, t), forn≥2, (1.2) with initial conditionsP0(s, t) = 0,P1(s, t) = 1 andQ0(s, t) = 2,Q1(s, t) = 2s.

In particular, ifs= 12, t= 1, then the classical Fibonacci and Lucas sequence are obtained, and ifs=t= 1, then the classical Pell and Pell-Lucas sequences are obtained. From the definition 1.1, we have that the characteristic equation of(1.1) and(1.2)are in the form

x2= 2sx+t (1.3)

and the root of equation (1.3) areα=s+√

s2+tandβ=s−√

s2+t. Note that α+β = 2s, α−β= 2√

s2+tand αβ=−t. Moreover, it can be seen that [1]

Qn(s, t) = 2sPn(s, t) + 2tPn−1(s, t), for alln≥0. (1.4) In this paper, we introduce the 2×2 matrix W which satisfy the relation W2= 2sW+tI.After that, we establish some identities of(s, t)-Pell and(s, t)-Pell- Lucas numbers and some sum formulas for(s, t)-Pell and(s, t)-Pell-Lucas numbers by using this matrix. Now, we first define (s, t)-Pell and(s, t)-Pell-Lucas numbers for negative subscript as follows:

Pn(s, t) =−Pn(s, t)

(−t)n , andQn(s, t) = Qn(s, t)

(−t)n , (1.5) for alln≥1. In the rest of this paper, for convenience we will use the symbolPn andQn instead ofPn(s, t)andQn(s, t)respectively.

2. Main results

We begin this section with the following Lemma.

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Lemma 2.1. If X is a square matrix withX2= 2sX+tI, then Xn=PnX+tPn−1I for all n∈N.

Proof. If n = 0, then the proof is obvious. It can be shown by induction that Xn=PnX+tPn−1Ifor alln∈N. Now, we will show thatXn=P−nX+tP−n−1I for alln∈N. LetY = 2sI−X =−tX1. Then we have

Y2= (2sI−X)2= 2s(2sI−X) +tI= 2sY +tI.

It implies thatYn =PnY +tPn1I. That is(−tX−1)n=Pn(2sI−X) +tPn1I.

Thus

(−t)nXn = 2sPnI− PnX+tPn1I

=−PnX+ (2sPn+tPn−1)I

=−PnX+Pn+1I.

Therefore,Xn=−(Pt)nnX+(Pn+1t)nI=P−nX+tP(n+1)I=P−nX+tP−n−1I.

This complete the proof.

By using Lemma 2.1, we obtain the Binet’s formula for (s, t)-Pell and (s, t)- Pell-Lucas numbers.

Corollary 2.2 (Binet’s formula). Thenth(s, t)-Pell and(s, t)-Pell-Lucas number are given by

Pn= αn−βn

α−β and Qnnn, for all n∈Z, where α = s+√

s2+t and β = s−√

s2+t are the roots of the characteristic equation (1.3).

Proof. Take X =

α 0 0 β

, then X2 = 2sX +tI. By Lemma 2.1, we have Xn=PnX+tPn1I. It follows that

αn 0 0 βn

=

αPn+tPn−1 0 0 βPn+tPn−1

.

Thus,αn=αPn+tPn1 andβn=βPn+tPn1, which implies that Pn= αn−βn

α−β and Qnnn, for all n∈Z. Let us define the2×2 matrixW as follows:

W =

s 2(s2+t)

1

2 s

. (2.1)

Then it easy to see thatW2= 2sW +tI. From this fact and Lemma 2.1, we get the following Lemma.

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Lemma 2.3. Let W be a matrix as in (2.1). Then Wn =

"1

2Qn 2(s2+t)Pn

1

2Pn 12Qn

#

for all n∈Z.

Proof. SinceW2= 2sW +tI, the proof follows from Lemma 2.1 and usingQn = 2sPn+ 2tPn1.

Now, by using the matrixW,we obtain some identities of(s, t)-Pell and (s,t)- Pell-Lucas numbers.

Lemma 2.4. Let m, n be any integers. Then the following results hold.

(i) Q2n−4(s2+t)Pn2= 4(−t)n, (ii) 2Qm+n =QmQn+ 4(s2+t)PmPn, (iii) 2Pm+n=PmQn+QmPn,

(iv) 2(−t)nQmn =QmQn−4(s2+t)PmPn, (v) 2(−t)nPm−n=PmQn− QmPn,

(vi) QmQn=Qm+n+ (−t)nQmn, (vii) PmQn=Pm+n+ (−t)nPmn.

Proof. Since det(Wn) = (det(W))n = (−t)n and det(Wn) = 14Q2n −(s2+t)Pn2, we get that Q2n −4(s2+t)Pn2 = 4(−t)n and then (i) immediately seen. Since Wm+n =WmWn, we obtain

" 1

2Qm+n 2(s2+t)Pm+n

1

2Pm+n 12Qm+n

#

=

" 1

4 QmQn+ 4(s2+t)PmPn

(s2+t)(QmPn+PmQn)

1

4(PmQn+QmPn) 14 4(s2+t)PmPn+QmQn

# . Thus, identities (ii) and (iii) are easily seen. Next, we note that Wmn = Wm(Wn) =Wm(Wn)1. Thus, we get that

" 1

2Qmn 2(s2+t)Pmn 1

2Pmn 1 2Qmn

#

= 1

(−t)n

" 1

4 QmQn−4(s2+t)PmPn

(s2+t)(−QmPn+PmQn)

1

4(PmQn− QmPn) 14 −4(s2+t)PmPn+QmQn

# . Therefore, the identities(iv)and(v)can be derived directly. The proof of(vi)and (vii)goes on in the same fashion as above by using the property

Wm+n+ (−t)nWmn=Wm(Wn+ (−t)nWn).

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Next, we give the following Lemma for using in the next Theorems.

Lemma 2.5. Let W be a matrix as in (2.1). Then H=W +tW−1=

0 4(s2+t)

1 0

.

Proof. Since det(W) = −t, we get that W1 =−1t

s −2(s2+t)

12 s

. Thus, H =

0 4(s2+t)

1 0

.

Finally, by using matricesW andH, we obtain some sum formulas for(s, t)-Pell and(s, t)-Pell-Lucas numbers.

Theorem 2.6. Let n∈Nandm, k∈Zwith (−t)m− Qm6=−1. Then Xn

j=0

Qmj+k =Qk− Qmn+m+k+ (−t)m Qmn+k− Qk−m 1 + (−t)m− Qm

and n

X

j=0

Pmj+k= Pk− Pmn+m+k+ (−t)m Pmn+k− Pkm 1 + (−t)m− Qm

Proof. It is know that

I−(Wm)n+1= (I−Wm) Xn j=0

(Wm)j.

By Lemma 2.4(i), we have det(I−Wm) = (1−1

2Qm)2−(s2+t)Pm2 = 1 + (−t)m− Qm. Sincedet(I−Wm)6= 0, we can write

(I−Wm)−1 I−(Wm)n+1 Wk=

Xn j=0

Wmj+k

=





 1 2

Xn j=0

Qmj+k 2(s2+t) Xn j=0

Pmj+k 1

2 Xn j=0

Pmj+k 1 2

Xn j=0

Qmj+k





. (2.2)

Since

(I−Wm)1= 1 1 + (−t)m− Qm

"

1−12Qm 2(s2+t)Pm

1

2Pm 1−12Qm

#

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= 1 1 + (−t)m− Qm

(1−1

2Qm)I+1 2PmH

,

we have

(I−Wm)−1 I−(Wm)n+1 Wk

=

(1−12Qm)I+12PmH

(Wk−Wmn+m+k) 1 + (−t)m− Qm

=

(1−12Qm)(Wk−Wmn+m+k) +12PmH(Wk−Wmn+m+k) 1 + (−t)m− Qm

= (1−1 2Qm)





1

2(Qk− Qmn+m+k) 1 + (−t)m− Qm

2(s2+t)(Pk− Pmn+m+k) 1 + (−t)m− Qm

1

2(Pk− Pmn+m+k) 1 + (−t)m− Qm

1

2(Qk− Qmn+m+k) 1 + (−t)m− Qm





+1 2Pm





2(s2+t)(Pk− Pmn+m+k) 1 + (−t)m− Qm

2(s2+t)(Qk− Qmn+m+k) 1 + (−t)m− Qm

1

2(Qk− Qmn+m+k) 1 + (−t)m− Qm

2(s2+t)(Pk− Pmn+m+k) 1 + (−t)m− Qm



 (2.3)

Using (2.2) and (2.3), we obtain Xn

j=0

Qmj+k

=

(1−12Qm)(Qk− Qmn+m+k) + 2(s2+t)Pm(Pk− Pmn+m+k)

1 + (−t)m− Qm . (2.4)

By Lemma 2.4(iv), (2.4) becomes Xn

j=0

Qmj+k =Qk− Qmn+m+k+ (−t)m Qmn+k− Qk−m 1 + (−t)m− Qm .

On the other hand, using (2.2) and (2.3) we get Xn

j=0

Pmj+k=

(1−12Qm)(Pk− Pmn+m+k) +12Pm(Qk− Qmn+m+k)

1 + (−t)m− Qm . (2.5) By Lemma 2.4(v), (2.5) becomes

Xn j=0

Pmj+k= Pk− Pmn+m+k+ (−t)m Pmn+k− Pk−m 1 + (−t)m− Qm .

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Theorem 2.7. Let n∈Nandm, k∈Zwith(−t)m+Qm6=−1. Ifnis even, then Xn

j=0

(−1)jQmj+k= Qk+Qmn+m+k+ (−t)m Qmn+k+Qk−m 1 + (−t)m+Qm

and n

X

j=0

(−1)jPmj+k =Pk+Pmn+m+k+ (−t)m Pmn+k+Pkm 1 + (−t)m+Qm

Proof. Letnis an even natural number. Then we have I+ (Wm)n+1= (I+Wm)

Xn j=0

(−1)j(Wm)j.

By Lemma 2.4(i), we have det(I+Wm) = (1 +1

2Qm)2−(s2+t)Pm2 = 1 +Qm+ (−t)m. Sincedet(I+Wm)6= 0, we can write

(I+Wm)−1 I+ (Wm)n+1 Wk

= Xn j=0

(−1)jWmj+k

=





 1 2

Xn j=0

(−1)jQmj+k 2(s2+t) Xn j=0

(−1)jPmj+k

1 2

Xn j=0

(−1)jPmj+k 1 2

Xn j=0

(−1)jQmj+k





. (2.6)

Since

(I+Wm)1= 1 1 +Qm+ (−t)m

 1 + 12Qm −2(s2+t)Pm

12Pm 1 + 12Qm

= 1

1 +Qm+ (−t)m

(1 +1

2Qm)I−1 2PmH

, we have

(I+Wm)−1 I+ (Wm)n+1 Wk

=

(1 +12Qm)I−12PmH

(Wk+Wmn+m+k) 1 +Qm+ (−t)m

=

(1 +12Qm)(Wk+Wmn+m+k)−12PmH(Wk+Wmn+m+k) 1 +Qm+ (−t)m

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= (1 +1 2Qm)



1

2(Qk+Qmn+m+k) 1 +Qm+ (−t)m

2(s2+t)(Pk+Pmn+m+k) 1 +Qm+ (−t)m

1

2(Pk+Pmn+m+k) 1 +Qm+ (−t)m

1

2(Qk+Qmn+m+k) 1 +Qm+ (−t)m



−1 2Pm



2(s2+t)(Pk+Pmn+m+k) 1 +Qm+ (−t)m

2(s2+t)(Qk+Qmn+m+k) 1 +Qm+ (−t)m

1

2(Qk+Qmn+m+k) 1 +Qm+ (−t)m

2(s2+t)(Pk+Pmn+m+k) 1 +Qm+ (−t)m



. (2.7)

Using (2.6) and (2.7), we obtain Xn

j=0

(−1)jQmj+k

=

(1 +12Qm)(Qk+Qmn+m+k)−2(s2+t)Pm(Pk+Pmn+m+k)

1 +Qm+ (−t)m . (2.8)

By Lemma 2.4(iv), (2.8) becomes Xn

j=0

(−1)jQmj+k = Qk+Qmn+m+k+ (−t)m Qkm+Qmn+k 1 + (−t)m+Qm . Similarly it can be easily seen that

Xn j=0

(−1)jPmj+k =Pk+Pmn+m+k+ (−t)m Pkm+Pmn+k 1 + (−t)m+Qm .

Theorem 2.8. Let n∈Nandm, k∈Zwith(−t)m+Qm6=−1. Ifnis odd, then Xn

j=0

(−1)jQmj+k= Qk− Qmn+m+k+ (−t)m Qkm− Qmn+k 1 + (−t)m+Qm

and n

X

j=0

(−1)jPmj+k =Pk− Pmn+m+k+ (−t)m Pkm− Pmn+k 1 + (−t)m+Qm

Proof. Letnis an odd natural number. Then we get Xn

j=0

(−1)jQmj+k=

n−1X

j=0

(−1)jQmj+k− Qmn+k.

Sincenis an odd natural number then n−1is even. By Thorem 2.7, we have

n1

X

j=0

(−1)jQmj+k =Qk+Qmn+k+ (−t)m Qmn+k−m+Qk−m 1 + (−t)m+Qm

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and Xn j=0

(−1)jQmj+k

= Qk+ (−t)m Qmn+km+Qkm

−(−t)mQmn+k− Qmn+kQm

1 + (−t)m+Qm . (2.9)

Using Lemma 2.4(vi)in (2.9), we get Xn

j=0

(−1)jQmj+k = Qk− Qmn+m+k+ (−t)m Qkm− Qmn+k 1 + (−t)m+Qm . In a similar way, it can be seen that

Xn j=0

(−1)jPmj+k=

n−1X

j=0

(−1)jPmj+k− Pmn+k.

By Theorem 2.7, it follows that Xn

j=0

(−1)jPmj+k

= Pk+ (−t)m Pmn+k−m+Pk−m

−(−t)mPmn+k− Pmn+kQm

1 + (−t)m+Qm . (2.10)

Using Lemma 2.4(vii)in (2.10), we obtain Xn

j=0

(−1)jPmj+k =Pk− Pmn+m+k+ (−t)m Pkm− Pmn+k 1 + (−t)m+Qm .

Acknowledgements. The authors would like to thank the faculty of science and technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thailand for the financial support. Moreover, the authors would like to thank the referees for their valuable suggestions and comments which helped to improve the quality and readability of the paper.

References

[1] H.H. Gulec, N. Taskara, On the(s, t)-Pell and(s, t)-Pell Lucas sequences and their matrix representations, Applied Mathematics Letters, Vol. 25 (2012), 1554–1559.

[2] T. Koshy, Pell and Pell-Lucas Numbers with Applications. Springer, Berlin (2014) [3] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons

Inc., New York (2001).

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[4] M.S. El Naschie, The Fibonacci Code behind Super Strings and P-Branes: An Answer to M. Kaku’s Fundamental Question. Chaos, Solitons & Fractals, Vol. 31 (2007), 537–547.

[5] M.S. El Naschie, Notes on Superstrings and the Infinite Sums of Fibonacci and Lucas Numbers. Chaos, Solitons & Fractals, Vol. 12 (2001), 1937–1940.

[6] A.P. Stakhov, Fibonacci Matrices: A Generalization of the “Cassini Formula” and a New Coding Theory. Chaos, Solitons Fractals, Vol. 30 (2006), 56–66.

[7] A.P. Stakhov, The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science and Engineering. Chaos, Solitons & Fractals, Vol. 26 (2005), 263–289.

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