Some Interesting Properties and Extended Binet Formula for the Generalized Lucas Sequence

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1 Some Interesting Properties and Extended Binet Formula for the Generalized ucas Sequence Daksha Diwan, Devbhadra V Shah 2 Assistant Professor, Department of Mathematics, Government Engineering College, Gandhinagar, Gujarat, India Associate Professor, Department of Mathematics, Sir PTSarvajanik College of Science, Surat,Gujarat, India 2 ABSTRACT: ucas sequence n is defined by the recurrence relation n n n 2 ; n 2with initial condition 0 2, One of the generalizations of the ucas sequence is the class of sequences n generated by the recurrence relation (a, b) n a (a, n b) n 2 ; when n is odd (a, b b) (a, n b) (n 2) n 2 ; when n is even with initial condition 0 2, and a, b are positive integers The ucas sequence is a special case of these sequences with a b In this paper we obtain extended Binet s formula for n Generalized ucas sequence KEYWORDS: ucas sequence, Generalized ucas sequence, Binet formula I INTRODUCTION and derive some interesting properties of this In recent years, many interesting properties of classic Fibonacci numbers, classic ucas number and their generalizations have been shown by researchers and applied to almost every field of science and art Cennet, Ahmet, Hasan [], Kenan, Adem [7] and Shah, Shah [] defined new generalizations of ucas sequence and gave various identities along with extended Binet formula for the concerned new generalizations For the rich and related applications of ucas numbers, one can refer to the nature and different areas of the science [2, 3, 4, 5, 6, 8, 3, 2, 4] The classic ucas sequence n is defined as 0 2, and n n n 2 for n 2 The first few terms of the ucas sequence are : 2,, 3, 4, 7,, 8, 29, 47, 76, 23, 99, 322, 52, 843, 364, 2207, 357 (Koshy [9]) We define a generalization of the ucas sequence and call it the generalized ucas sequence Definition: For any two positive numbers a and b, the generalized ucas sequence G n (a, b,,) n n is defined by the recurrence relation n a χ n b χ n n where 0 2, and a, b are any two positive integers This can be equivalently expressed as n a n n 2 ; if n is odd (n 2) () b n n 2 ; if n is even with 0 2, n 2, Copyright to IJIRSET DOI:05680/IJIRSET

2 The ucas sequence is a special case of these sequences with a b In this paper we derive extended Binet s formula for n and some of its interesting properties IIEXTENDED BINET S FORMUA FOR n We first note down two results which will be useful for further manipulations emma 2: 2n5 2 2n3 Proof: By using definition (), we get 2n5 2 2n3 2n (a 2n2 2n ) 2n3 0 2n 0 2n3 a 2n2 We now obtain the value of the following series related with n 2 2n3 2n emma 22: i 2i x 2i x(2a )x 3 Proof: et l x i 2i x 2i x 3 x 3 5 x 5 x 4 l x x 5 3 x 7 5 x 9 and ( 2)x 2 l x x 5 3 x 7 5 x 9 Using lemma 2, we get (x 4 2 x 2 )l(x) x (2a )x 3 This gives l x i 2i x 2i x(2a )x 3 We next derive the generating function for n emma 23: The generating function for n is given by x (2a )x b x 2 x2 Proof: Define x i0 i x i 0 x 2 x 2 3 x 3 Then after the rearrangement of terms, we get ax x 2 x 2 2a x b a x i 2i x 2i Using emma 22 and on simplification we get the generating function for n as ( 2a x b x 2 x2) x 2 ( 2a)x ( ax x 2 ) b a x ( ax x 2 ) x (2a )x 3 x 4 2 x 2 ( 2a x3 2 2 b x 2 x 2) ax x 2 ax x 2 x 4 2 x 2 We are now all set to derive the extended Binet s formula for n Theorem 24: n γ n 2 bχ n n δβ n δ b 2 2β and a2 b 2 4 2, where γ b 2 2, Proof: By lemma 23, we have x ( 2a x b x 2 x2) If we write x ( 2a x b x 2 x2) we get, β a2 b 2 4 be the roots of equation x 2 x 0 2 (Ax B) (CxD) x 2 () x 2 (β ), then by using the method of partial fractions, Copyright to IJIRSET DOI:05680/IJIRSET

3 2a x2 ( (2 2 b) x () x 2 () Then by using Maclaurin s expansion [0] of P Qx as x 2 R we get and Thus, ( ) P Qx x 2 R 2a β x2 (2 2 b)(β ) QR n x 2n x 2 (β) PR n x 2n b 2a x x 2 2a n x 2n 2a β β n x 2n x, 2 β 2 2 b 2a β x x 2 β (2 β 2 2 b ) β n x 2n 2 (β)(2 2 b) β n 2 ()(22 b) () n 2a n x 2n Now using 2 β 2 and β we get (x) () 2a 2a b a n β β 2n 2n β 2n β 2n 2a n 2n 2 β 2n 2 x 2n () β 2n 2 Using β, we get (x) () β (2 2 b) ( ) n x 2n 2a β β n x 2n n β 2 n β 2 n β 2 n 2 n 2 n β 2 n β 2 2 n 2 n β 2 β 2 n 2 n 2 n β 2 n β 2 n 2 n 2a n β 2n 2n 2n b 2 2 2n β 2n () n n x 2n x 2n b 2 2 n β 2 2n β 2n β 2 2n β 2n 2n 2n 2 2 n 2n 2 β x β 2n 2 2a 2n2 β 2n2 n x 2n 2 2n2 β 2n2 n x 2n Copyright to IJIRSET DOI:05680/IJIRSET

4 () x 2n () n Using 2 β 2 and β we get ( ) x x 2n b n b 2 2 2n b 2 2β β 2n b x 2n b n () x 2n b n 2n b β2 b 2 2 2n b 2 2β β 2n x 2n () n x 2n (b 2 ( ) n 2)2n (b 2 2β)β 2n Thus if we write γ b 2 2 and δ b 2 2β, then x b n b β β 2n b 2 2 2n b 2 2β β 2n γ n δβ n n x n 2 bχ n β 2n Hence the required extended Binet formula is n where γ b 2 2, δ b 2 2β γn n 2 bχ n δβn IIISOME INTERESTING PROPERTIES OF n Here we derive some interesting properties for n from its extended Binet formula We first find expressions which characterizes n in terms of n and powers of or β Theorem 3: n a χ n b χ n n β n ( ) n 2 a χ n ; n 2 Proof: By theorem 24 we have n C γ n δβ n, where C, γ b 2 2, δ b 2 2β b χ n n 2 Now consider the cases n 2kand n 2k 2k b 2k 2k 2k 2 2k b 2k γ 2k δβ 2k k ( ) δβ2k b( ) k Thus γ 2k δβ 2k ( ) k and 2k γ 2k δβ 2k b( ) k γ 2k δβ 2k γ 2k δβ 2k δβ2k ( ) ( ) k () 2k b 2k δβ2k b( ) k (3) Again considering n 2k and n 2k 2 in theorem 24, we get 2k b 2k 2 γ 2k δβ 2k Using these two results, we get γ 2k δβ 2k b( ) k and 2k2 γ 2k2 δβ 2k2 ( ) k Copyright to IJIRSET DOI:05680/IJIRSET

5 2k2 2k2 2k2 b 2k a 2k a 2k k ( ) γ2k2 δβ 2k2 γ 2k2 δβ 2k δβ 2k ( ) ( ) k ( ) δβ 2k ( ) δβ 2k k This gives ( ) k (32) Finally, combining (3) and (32), we get Theorem 32 : n β a χ n b χ n n a χ n b χ n n δβn ( ) n 2 bχ n n n ( ) n 2 b χ n ; n 2 Proof: From theorem 24 we have, n C γ n δβ n, where C γ b 2 2, δ b 2 2β Now consider the cases when n 2kor n 2k 2k 2k 2 γ 2k δβ 2k Using these two results we get b 2k 2k 2k β 2k β b 2k β b 2k γ 2k δβ 2k ( ) k b χ n n 2, and 2k γ 2k δβ 2k ( ) k k ( ) γ2k δβ 2k γ 2k β δβ 2k γ2k ( ) ( ) k ( ) γ 2k b( ) γ 2k k Thus b( ) k (33) Again considering n 2k and n 2k 2 in theorem 24, we get 2k b 2k 2 γ 2k δβ 2k γ 2k δβ 2k b( ) k and 2k2 γ 2k2 δβ 2k2 ( ) k 2k2 bβ 2k k ( ) γ2k2 δβ 2k2 γ 2k β δβ 2k2 γ 2k ( ) 2k2 2k2 β a 2k β a 2k γ 2k ( ) γ 2k k Hence ( ) k(34) Combining (33) and (34) we get Corollary 33:2 n β a χ n b χ n n n γn δβ n ( ) n 2 bχ n Proof: Adding the results of theorems 3and 32, we get 2 n β a χ n b χ n n β a χ n b χ n n ( ) k ( ) γn χ n ( ) n 2 b γn δβ n ( ) n, as required 2 bχ n IVCONCUSION In this paper we derived extended Binet s formula for n this Generalized ucas sequence in section 3 in theorem 24 and derive some interesting properties of REFERENCES [] CennetBolat, AhmetIpek, HasanKose : On the sequence related to ucas numbers and its properties, MathematicaAeterna,Vol 2, No, 63 75,203 [2] Civcic H and Turkmen R :Notes on the (s,t)-lucas and ucas Matrix sequence, ArsCombinatoria, 89, , (2008) [3] Falcon S, Plaza, A :On the Fibonacci K-numbers, Chaos, Solitons& Fractals, 32 (5), 65 24,(2007) Copyright to IJIRSET DOI:05680/IJIRSET

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