On the (s,t)-pell and (s,t)-pell-lucas numbers by matrix methods

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1 Annales Mathematicae et Informaticae pp 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 0,Thailand Submitted March 7, 06 Accepted September 5, 06 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 the matrix W, which satisfy the relation W sw + 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 number; s, t-pell number; s, t-pell-lucas number MSC: B37; 5A5 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:, 3, 4, 5, 6, 7 For n, the classical Fibonacci {F n }, Lucas {L n }, Pell {P n } and Pell-Lucas {Q n } This research was supported by faculty of science and technology, Rajamangala University of Technology Thanyaburi RMUTT, Thailand 95

2 96 S Srisawat, W Sriprad sequences are defined by 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, with the initial conditions F 0 0, F, L 0, L, P 0 0, P and Q 0 Q, respectively For more detialed information about Fibonacci, Lucas, Pell, Pell-Lucas sequences can be found in, 3 Recently, Fibonacci, Lucas, Pell and Pell-Lucas were generalized and studied by many authors in the different ways to derive many identities In 0, Gulec and Taskara 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 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 Let s, t be any real number with s + t > 0, s > 0 and t 0 Then the s, t-pell sequences {P n s, t} n N and the s, t-pell-lucas sequences {Q n s, t} n N are defined respectively by P n s, t sp n s, t + tp n s, t, for n, Q n s, t sq n s, t + tq n s, t, for n, with initial conditions P 0 s, t 0, P s, t and Q 0 s, t, Q s, t s In particular, if s, t, then the classical Fibonacci and Lucas sequence are obtained, and if s t, then the classical Pell and Pell-Lucas sequences are obtained From the definition, we have that the characteristic equation of and are in the form x sx + t 3 and the root of equation 3 are α s + s + t and β s s + t Note that α + β s, α β s + t and αβ t Moreover, it can be seen that Q n s, t sp n s, t + tp n s, t, for all n 0 4 In this paper, we introduce the matrix W which satisfy the relation W sw +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: P n s, t P ns, t t n, and Q ns, t Q ns, t t n, 5 for all n In the rest of this paper, for convenience we will use the symbol P n and Q n instead of P n s, t and Q n s, t respectively Main results We begin this section with the following Lemma

3 On the s,t-pell and s,t-pell-lucas numbers by matrix methods 97 Lemma If X is a square matrix with X sx + ti, then X n P n X + tp n I for all n N Proof If n 0, then the proof is obvious It can be shown by induction that X n P n X+tP n I for all n N Now, we will show that X n P n X+tP n I for all n N Let Y si X tx Then we have Y si X ssi X + ti sy + ti It implies that Y n P n Y + tp n I That is tx n P n si X + tp n I Thus t n X n sp n I P n X + tp n I P n X + sp n + tp n I P n X + P n+ I Therefore, X n Pn t n X + Pn+ t n I P n X + tp n+ I P n X + tp n I This complete the proof By using Lemma, we obtain the Binet s formula for s, t-pell and s, t- Pell-Lucas numbers Corollary Binet s formula The n th s, t-pell and s, t-pell-lucas number are given by P n αn β n α β and Q n α n + β n, for all n Z, where α s + s + t and β s s + t are the roots of the characteristic equation 3 α 0 Proof Take X, then X 0 β sx + ti By Lemma, we have X n P n X + tp n I It follows that α n 0 αpn + tp 0 β n n 0 0 βp n + tp n Thus, α n αp n + tp n and β n βp n + tp n, which implies that P n αn β n α β and Q n α n + β n, for all n Z Let us define the matrix W as follows: s s W + t s Then it easy to see that W sw + ti From this fact and Lemma, we get the following Lemma

4 98 S Srisawat, W Sriprad Lemma 3 Let W be a matrix as in Then W n for all n Z Q n s + tp n P n Q n Proof Since W sw + ti, the proof follows from Lemma and using Q n sp n + tp n Now, by using the matrix W, we obtain some identities of s, t-pell and s,t- Pell-Lucas numbers Lemma 4 Let m, n be any integers Then the following results hold i Q n 4s + tp n 4 t n, ii Q m+n Q m Q n + 4s + tp m P n, iii P m+n P m Q n + Q m P n, iv t n Q m n Q m Q n 4s + tp m P n, v t n P m n P m Q n Q m P n, vi Q m Q n Q m+n + t n Q m n, vii P m Q n P m+n + t n P m n Proof Since detw n detw n t n and detw n 4 Q n s + tpn, we get that Q n 4s + tpn 4 t n and then i immediately seen Since W m+n W m W n, we obtain Q m+n s + tp m+n P m+n Q m+n 4 Qm Q n + 4s + tp m P n s + tq m P n + P m Q n 4 P mq n + Q m P n 4 4s + tp m P n + Q m Q n Thus, identities ii and iii are easily seen Next, we note that W m n W m W n W m W n Thus, we get that Q m n s + tp m n P m n t n Q m n 4 Qm Q n 4s + tp m P n s + t Q m P n + P m Q n 4 P mq n Q m P n 4 4s + tp m P n + Q m Q n 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 W m+n + t n W m n W m W n + t n W n

5 On the s,t-pell and s,t-pell-lucas numbers by matrix methods 99 Next, we give the following Lemma for using in the next Theorems Lemma 5 Let W be a matrix as in Then 0 4s H W + tw + t 0 s s Proof Since detw t, we get that W + t t s 0 4s H + t 0 Thus, Finally, by using matrices W and H, we obtain some sum formulas for s, t-pell and s, t-pell-lucas numbers Theorem 6 Let n N and m, k Z with t m Q m Then Q mj+k Q k Q mn+m+k + t m Q mn+k Q k m + t m Q m and j0 j0 Proof It is know that By Lemma 4 i, we have P mj+k P k P mn+m+k + t m P mn+k P k m + t m Q m I W m n+ I W m W m j j0 deti W m Q m s + tp m + t m Q m Since deti W m 0, we can write I W m I W m n+ W k j0 W mj+k Q mj+k s + t j0 j0 P mj+k j0 j0 Q mj+k P mj+k Since I W m + t m Q m Q m s + tp m P m Q m

6 00 S Srisawat, W Sriprad we have + t m Q m Q mi + P mh, I W m I W m n+ W k Q mi + P mh W k W mn+m+k + t m Q m Q mw k W mn+m+k + P mhw k W mn+m+k + t m Q m Q k Q mn+m+k s + tp k P mn+m+k + t m Q m + t m Q m Q m + P m P k P mn+m+k + t m Q m Q k Q mn+m+k + t m Q m s + tp k P mn+m+k + t m Q m s + tq k Q mn+m+k + t m Q m Q k Q mn+m+k + t m Q m s + tp k P mn+m+k + t m Q m 3 Using and 3, we obtain j0 Q mj+k Q mq k Q mn+m+k + s + tp m P k P mn+m+k + t m 4 Q m By Lemma 4 iv, 4 becomes j0 Q mj+k Q k Q mn+m+k + t m Q mn+k Q k m + t m Q m On the other hand, using and 3 we get Q mp k P mn+m+k + P mq k Q mn+m+k P mj+k + t m 5 Q m j0 By Lemma 4 v, 5 becomes j0 P mj+k P k P mn+m+k + t m P mn+k P k m + t m Q m

7 On the s,t-pell and s,t-pell-lucas numbers by matrix methods 0 Theorem 7 Let n N and m, k Z with t m + Q m If n is even, then j Q mj+k Q k + Q mn+m+k + t m Q mn+k + Q k m + t m + Q m and j0 j P mj+k P k + P mn+m+k + t m P mn+k + P k m + t m + Q m j0 Proof Let n is an even natural number Then we have I + W m n+ I + W m j W m j By Lemma 4 i, we have deti + W m + Q m s + tpm + Q m + t m Since deti + W m 0, we can write j0 I + W m I + W m n+ W k j W mj+k j0 j Q mj+k s + t j P mj+k j0 j P mj+k j0 j0 j Q mj+k j0 6 Since we have I + W m + Q m s + tp m + Q m + t m P m + Q m + Q m + t m + Q mi P mh, I + W m I + W m n+ W k + Q mi P mh W k + W mn+m+k + Q m + t m + Q mw k + W mn+m+k P mhw k + W mn+m+k + Q m + t m

8 0 S Srisawat, W Sriprad + Q m Q k + Q mn+m+k s + tp k + P mn+m+k + Q m + t m + Q m + t m P k + P mn+m+k + Q m + t m Q k + Q mn+m+k + Q m + t m s + tp k + P mn+m+k s + tq k + Q mn+m+k P + Q m + t m + Q m + t m m Q k + Q mn+m+k s + tp k + P mn+m+k + Q m + t m + Q m + t m Using 6 and 7, we obtain j Q mj+k j0 7 + Q mq k + Q mn+m+k s + tp m P k + P mn+m+k + Q m + t m 8 By Lemma 4 iv, 8 becomes j Q mj+k Q k + Q mn+m+k + t m Q k m + Q mn+k + t m + Q m j0 Similarly it can be easily seen that j P mj+k P k + P mn+m+k + t m P k m + P mn+k + t m + Q m j0 Theorem 8 Let n N and m, k Z with t m + Q m If n is odd, then j Q mj+k Q k Q mn+m+k + t m Q k m Q mn+k + t m + Q m and j0 j P mj+k P k P mn+m+k + t m P k m P mn+k + t m + Q m j0 Proof Let n is an odd natural number Then we get n j Q mj+k j Q mj+k Q mn+k j0 j0 Since n is an odd natural number then n is even By Thorem 7, we have n j Q mj+k Q k + Q mn+k + t m Q mn+k m + Q k m + t m + Q m j0

9 On the s,t-pell and s,t-pell-lucas numbers by matrix methods 03 and j Q mj+k j0 Q k + t m Q mn+k m + Q k m t m Q mn+k Q mn+k Q m + t m + Q m 9 Using Lemma 4 vi in 9, we get j Q mj+k Q k Q mn+m+k + t m Q k m Q mn+k + t m + Q m j0 In a similar way, it can be seen that n j P mj+k j P mj+k P mn+k j0 By Theorem 7, it follows that j P mj+k j0 j0 P k + t m P mn+k m + P k m t m P mn+k P mn+k Q m + t m + Q m 0 Using Lemma 4 vii in 0, we obtain j P mj+k P k P mn+m+k + t m P k m P mn+k + t m + Q m j0 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 HH Gulec, N Taskara, On the s, t-pell and s, t-pell Lucas sequences and their matrix representations, Applied Mathematics Letters, Vol 5 0, T Koshy, Pell and Pell-Lucas Numbers with Applications Springer, Berlin 04 3 T Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, New York 00

10 04 S Srisawat, W Sriprad 4 MS El Naschie, The Fibonacci Code behind Super Strings and P-Branes: An Answer to M Kaku s Fundamental Question Chaos, Solitons & Fractals, Vol 3 007, MS El Naschie, Notes on Superstrings and the Infinite Sums of Fibonacci and Lucas Numbers Chaos, Solitons & Fractals, Vol 00, AP Stakhov, Fibonacci Matrices: A Generalization of the Cassini Formula and a New Coding Theory Chaos, Solitons Fractals, Vol , AP Stakhov, The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science and Engineering Chaos, Solitons & Fractals, Vol 6 005, 63 89