Applied Mathematics Letters

Transcription

1 Applied Mathematics Letters 5 (0) Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: wwwelseviercom/locate/aml On the (s, t)-pell and (s, t)-pell Lucas sequences and their matrix representations Hasan Huseyin Gulec, Necati Taskara Selcuk University, Science Faculty, Department of Mathematics, 4075, Campus, Konya, Turkey a r t i c l e i n f o a b s t r a c t Article history: Received 6 October 0 Accepted 8 January 0 Keywords: Pell numbers Pell Lucas numbers Matrix representations In this paper, we first give new generalizations for (s, t)-pell {p n (s, t)} n N and (s, t)- Pell Lucas {q n (s, t)} n N sequences for Pell and Pell Lucas numbers Considering these sequences, we define the matrix sequences which have elements of {p n (s, t)} n N and {q n (s, t)} n N Then we investigate their properties 0 Elsevier Ltd All rights reserved Introduction Fibonacci, Lucas, Pell, and Pell Lucas sequences have been discussed in many articles and books (see [ 4]) For n >, the well-known Fibonacci {F n }, Lucas {L n }, Pell {p n }, and Pell Lucas {q n } sequences are defined as F n F n + F n, L n L n + L n, p n p n + p n, and q n q n + q n, where F 0 0, F, L 0, L, p 0 0, p, and q 0, q The closed-form expressions for the Fibonacci and Lucas numbers are F n αn β n and L n α n + β n, where α + 5 and β 5 Also, for a + and b, the Pell number and Pell Lucas number are p n an b n a b and q n a n + b n, where a and b are the roots of the equation x x+ Further details about Pell and Pell Lucas numbers can be found in [5] In [6], Kılıç gave the definition of generalized Pell (p, i)-numbers and then presented their generating matrix He obtained relationships between the generalized Pell (p, i)-numbers and their sums and permanents of certain matrices Also, he derived the generalized Binet formulas, sums, combinatorial representations In [7,8], the authors defined a new matrix generalization of the Fibonacci and Lucas numbers, and using essentially a matrix approach they showed properties of these matrix sequences The (s, t)-pell, (s, t)-pell Lucas sequences and their matrix sequences In this section, we give definitions of the (s, t)-pell and (s, t)-pell Lucas sequences and (s, t)-pell and (s, t)-pell Lucas matrix sequences We also investigate their properties Definition For any real numbers s, t and n, let s + t > 0, s > 0 and t 0 Then the (s, t)-pell sequence {p n (s, t)} n N and the (s, t)-pell Lucas sequence {q n (s, t)} n N are defined respectively by p n (s, t) sp n (s, t) + tp n (s, t), q n (s, t) sq n (s, t) + tq n (s, t), with initial conditions p 0 (s, t) 0, p (s, t) and q 0 (s, t), q (s, t) s α β () () Corresponding author addresses: hhgulec@selcukedutr (HH Gulec), ntaskara@selcukedutr (N Taskara) /$ see front matter 0 Elsevier Ltd All rights reserved doi:006/jaml0004

2 HH Gulec, N Taskara / Applied Mathematics Letters 5 (0) Thus one can obtain the characteristic equation of () and () in the form x sx+t; then the roots of the characteristic equation of () and () are x s+ s + t and x s s + t Note that x +x s, x x s + t and x x t For some special values of s and t in (), it is obvious that the following results hold If s, t, the classic Fibonacci sequence is obtained If s t, the classic Pell sequence is obtained If s, t, the classic Jacobsthal sequence is obtained If s 3, t, the Mersenne sequence is obtained Also, for some special values of s and t in (), it is obvious that the following results hold If s, t, the classic Lucas sequence is obtained If s t, the classic Pell Lucas sequence is obtained If s, t, the classic Jacobsthal Lucas sequence is obtained If s 3, t, the Fermat sequence is obtained Let us consider the following proposition, which will be needed for the results in this section In fact, by this proposition, there will be given a relationship between the sequences {p n (s, t)} n N and {q n (s, t)} n N Proposition For n 0, we have q n (s, t) sp n (s, t) + tp n (s, t) (3) Proof To prove the existence of this equality, we need to consider the sequence given in () with its initial conditions If we consider the initial condition q 0 (s, t), then the expression can be written as q 0 (s, t) (s) 0 + (t) t If we apply same idea to the other condition q (s, t) s, then we have q (s, t) s (s) + (t) 0 In fact, these rewritten conditions contain the initial conditions p (s, t), p 0 (s, t) and p (s, t) of the (s, t)-pell sequence Therefore, by replacing these conditions by these new q 0 (s, t) and q (s, t), we obtain q 0 (s, t) (s) p 0 (s, t) + (t) p (s, t), q (s, t) s (s) p (s, t) + (t) p 0 (s, t) By keeping the (s, t)-pell sequence and using same technique, we get (s) p (s, t) + (t) p (s, t), which gives q (s, t) in the statement of proposition So, by iterating process, we obtain the general term in the sp n (s, t) + tp n (s, t), which implies q n (s, t), as required In the following proposition, using the same approximation as in Proposition, we will show that there are also some other relations between {p n (s, t)} n N and {q n (s, t)} n N without any proof Proposition 3 For n 0, we have q n+ (s, t) + tq n+ (s, t) 4(s + t)p n+3 (s, t), q n+ (s, t) + tq n+ (s, t) q n+4(s, t) + tq n+ (s, t), q n (s, t) p n (s, t)q n+ (s, t) + tp n (s, t)q n (s, t) Now, considering these sequences, we define the matrix sequences which have elements of (s, t)-pell and (s, t)- Pell Lucas sequences Definition 4 Let s, t R, s > 0, t 0, s + t > 0, and n The (s, t)-pell matrix sequence {P n (s, t)} n N and (s, t)-pell Lucas matrix sequence {Q n (s, t)} n N are defined respectively by P n (s, t) sp n (s, t) + tp n (s, t), Q n (s, t) sq n (s, t) + tq n (s, t), with initial conditions P 0 (s, t), P (s, t) 0 0 s, and Q t 0 0 (s, t) s t s, Q (s, t) 4s + t st In the rest of this paper, the (s, t)-pell and (s, t)-pell Lucas matrix sequences will be denoted by P n and Q n instead of P n (s, t) and Q n (s, t), respectively s t form (4) (5)

3 556 HH Gulec, N Taskara / Applied Mathematics Letters 5 (0) Theorem 5 For n 0, we have (a) P n P x P 0 x n P x P 0 x x x n x x, (b) Q n Q x Q 0 x n Q x Q 0 x n x x x x Proof (a) The solution of Eq (4) is P n c x n + c x n (6) Then, let P 0 c + c, P c x + c x Therefore, we have c P x P 0 x x, c P x P 0 Using c x x and c in Eq (6), we obtain P x P 0 P n x n x x P x P 0 x n x x (b) The proof is similar to the proof of (a) The following theorem gives us the nth general term of the sequence given in (4) and (5) Theorem 6 For n 0, we have (a) P n pn+ (s, t) p n (s, t) tp n (s, t) tp n (s, t), (b) Q n qn+ (s, t) q n (s, t) tq n (s, t) tq n (s, t) Proof (a) Let use the principle of mathematical induction on n Let us consider n 0 in () We have p (s, t), p 0 (s, t) 0 and p (s, t) Then we write t p (s, t) p P 0 0 (s, t) 0 tp 0 (s, t) tp (s, t) 0 By iterating this procedure and considering induction steps, let us assume that the equality in (a) holds for all n k Z + To finish the proof, we have to show that (a) also holds for n k + by considering () and (4) Therefore we get P k+ sp k + tp k spk+ (s, t) + tp k (s, t) sp k (s, t) + tp k (s, t) stp k (s, t) + t p k (s, t) stp k (s, t) + t p k (s, t) pk+ (s, t) p k+ (s, t) tp k+ (s, t) tp k (s, t) Hence we obtain the result If a similar argument is applied to (b), the proof is clearly seen Theorem 7 Assume that s + t > 0, s > 0 and t 0 We obtain n P (a) k k0 xp + (x sx)p x k x sx t 0 x n (x sx t) (xp n+ + tp n ), n Q (b) k k0 xq + (x sx)q x k x sx t 0 x n (x sx t) (xq n+ + tq n ) Proof In contrast, here we will just prove (b) since the proof of (a) can be done in a similar way From Theorem 5, we have n Q k x Q x Q 0 n x k Q x Q 0 n x k k x x x x x x k0 k0 By considering the definition of a geometric sequence, we get n Q k Q x Q 0 x n+ x n+ x k x k0 Q x Q 0 x n+ x n+ x x n+ x x x x x x n+ x x x Q x Q 0 x n+ x n+ Q x Q 0 x x n (x (x x ) n+ x n+ (x x ) sx t) x x x x If we rearrange the last equality, then we obtain n Q k x xq + (x sx)q k x 0 sx t x n (x sx t) (xq n+ + tq n ) k0 k0

4 HH Gulec, N Taskara / Applied Mathematics Letters 5 (0) In the following theorem, we give the sums of (s, t)-pell and (s, t)-pell Lucas matrix sequences corresponding to different indices Theorem 8 For j > m, we have n (a) P i0 mi+j P mn+m+j+( t) m P j m ( t) m P mn+j P j x m, +xm ( t)m n (b) Q i0 mi+j Q mn+m+j+( t) m Q j m ( t) m Q mn+j Q j x m +xm ( t)m Proof (a) Let us take A P x P 0, B P x P 0 Then, we write x x n P mi+j i0 n i0 x x Ax mi+j Bx mi+j x x x x x x n Ax j i0 Ax j x mi Bxj x m(n+) x m n i0 x mi Bx j After some algebra, we obtain n P mi+j P mn+m+j + ( t) m P j m ( t) m P mn+j P j x m + xm ( t)m i0 (b) The proof is similar to the proof of (a) x m(n+) x m 3 The relationships between matrix sequences P n and Q n Proposition 9 For m, n Z +, the (s, t)-pell and (s, t)-pell Lucas matrix sequences are commutative The following results hold P m P n P n P m P m+n, Q m Q n Q n Q m, Q P n P n Q Q n+, Q n P P Q n Q n+, P n Q n+ Q n+ Proof The proof can be seen clearly by using Theorem 6 and a matrix multiplication or induction method Theorem 0 For m, n Z + the following properties hold (a) Q n sp n + tp n, (b) Q n P n+ + tp n, (c) Q m Q n (4s + 4t)P m+n, (d) Q P n P n+ + tp n, (e) P n P m sq m+n + t P m+n 4, (m + n 4) Proof First, here, we will just prove (a) and (d) since (b), (c), and (e) can be dealt with in the same manner So, if we consider the right-hand side of equation (a) and use Theorem 6, we get pn+ (s, t) p sp n + tp n s n (s, t) pn (s, t) p + t n (s, t) tp n (s, t) tp n (s, t) tp n (s, t) tp n (s, t) From Eq (3), sp n + tp n qn+ (s, t) q n (s, t) tq n (s, t) tq n (s, t) Q n, as required in (a) Second, let us consider the left-hand side of equation (d) Using Theorem 6, we write 4s Q P n + t s pn+ (s, t) p n (s, t) st t tp n (s, t) tp n (s, t) From matrix production, we have pn+3 (s, t) p Q P n n+ (s, t) + t tp n+ (s, t) tp n+ (s, t) Hence the result P n+ + tp n pn+ (s, t) p n (s, t) tp n (s, t) tp n (s, t)

5 558 HH Gulec, N Taskara / Applied Mathematics Letters 5 (0) Theorem For m, n Z + the following properties hold (a) P m Q n+ Q n+ P m Q m+n+, (b) Q m n+ Qm P mn Proof (a) Let us consider the left-hand side of equation (a) and Proposition 9 We have P m Q n+ P m Q P n Then, by Theorem 0(a), we get P m Q n+ P m (sp + tp 0 )P n Now, by considering Proposition 9, we write P m Q n+ sp n+m+ + tp n+m sp P n+m + tp n+m (sp + tp 0 )P n+m Moreover, from Theorem 0(a), we obtain P m Q n+ Q P n P m Q n+ P m Also, from Proposition 9, it is seen that P m Q n+ Q m+n+ which finishes the proof of (a) (b) To prove equation (b), let us follow induction steps on m For m, the proof is clear by Proposition 9 Now, assume that it is true for all positive integers m, that is, Q m n+ Qm P mn Therefore, we have to show that it is true for m + If we multiply this mth step by Q n+ on both sides from the right, then we have Q m+ n+ Qm P mnq n+ Also, by considering Proposition 9 and Theorem (a), we write Q m+ n+ Qm+ P n P mn Q m+ P (m+)n which finishes the induction and gives the proof of (b) Corollary For n 0, by taking m in the equation given in Theorem (b), we obtain Q n+ Q P n Q Q n+ Theorem 3 For m, n 0 and n r, the following results hold (a) P n r P n+r P n, (b) P m P n+ P m+ P n, (c) Q n r Q n+r Q n, (d) Q m Q n+ Q m+ Q n Proof In the proof, we will consider only the conditions (a) and (d), since the others can be dealt with similarly (a) Let us take A P x P 0, B P x P 0 Then, using Theorem 5, we write Ax n r P n r P n+r P Bx n r Ax n+r Bx n+r Ax n n Bxn x x x x x x If we rearrange the last equation, we have P n r P n+r P n have P n r P n+r P n, as required (d) From Theorem 0(c), we obtain Q m Q n+ (4s + 4t)P m+n+ Q m+ Q n ABxn r x n r (x r xr xr xr ) Consequently, from A B [0] (x x ) x, we Acknowledgment This research is supported by TUBITAK and Selcuk University Scientific Research Project Coordinatorship (BAP) This study forms part of the corresponding author s PhD Thesis

6 HH Gulec, N Taskara / Applied Mathematics Letters 5 (0) References [] AT Benjamin, SS Plott, JA Sellers, Tiling proofs of recent sum identities involving Pell numbers, Annals of Combinatorics (008) 7 78 [] N Taskara, K Uslu, HH Gulec, On the properties of Lucas numbers with binomial coefficients, Applied Mathematics Letters 3 () (00) 68 7 [3] A Stakhov, B Rozin, Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos, Solitons & Fractals 7 (5) (006) 6 77 [4] T Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 00 [5] M Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quarterly 3 (975) [6] E Kılıç, The generalized Pell (p, i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos, Solitons & Fractals 40 (009) [7] H Civciv, R Türkmen, On the (s, t)-fibonacci and Fibonacci matrix sequences, Ars Combinatoria 87 (008) 6 73 [8] H Civciv, R Türkmen, Notes on the (s, t)-lucas and Lucas matrix sequences, Ars Combinatoria 89 (008) 7 85