Generalized Fibonacci-Like Polynomial and its. Determinantal Identities

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1 Int. J. Contemp. Math. Scences, Vol. 7, 01, no. 9, Generalzed Fbonacc-Le Polynomal and ts Determnantal Identtes V. K. Gupta 1, Yashwant K. Panwar and Ompraash Shwal 3 1 Department of Mathematcs, Govt. Madhav Scence College, Uan, Inda dr_vg61@yahoo.com Department of Mathematcs Mandsaur Insttute of Technology,Mandsaur, Inda yashwantpanwar@gmal.com 3 Department of Mathematcs Mandsaur Insttute of Technology,Mandsaur, Inda opbhshwal@redffmal.com, opshwal@gmal.com Abstract It s well nown that the Fbonacc polynomals are of great mportance n the study of many subects such as Algebra, geometry, combnatorcs and number theory tself. Fbonacc polynomals defned by the recurrence relaton fn() fn 1() fn (), n wth f 0 () 0, f 1 () 1. In ths paper we ntroduce Generalzed Fbonacc-Le Polynomals. Further we present ts generalzed determnantal denttes wth classcal polynomals le Fbonacc Polynomal, Lucas Polynomals, Pell Polynomals and Pell-Lucas Polynomals. Mathematcs Subect Classfcaton: 11B39, 11B37, 11C08, 11C0 Keywords: Fbonacc polynomal, Fbonacc-Le polynomal, Determnant 1. INTRODUCTION Fbonacc polynomals defned by the recurrence relaton fn() fn 1() fn (), n wth f0() 0, f1() 1. It s well nown that

2 1416 V. K. Gupta, Y. K. Panwar and O. Shwal the Fbonacc polynomals are of great mportance n the study of many subects such as Algebra, geometry, combnatorcs and number theory tself. Many authors have studed Fbonacc polynomals and Generalzed Fbonacc polynomals denttes. They appled concept of Matr and Determnants to establsh some denttes. Spvey [8] descrbe sum property for determnants and presented new proof denttes le Cassn dentty, d Ocagne dentty and Catalan dentty. Koen and Bozurt [6] defne Jacobsthal M-matr and Jacobsthal Q- matr smlar to Fbonacc Q-matr and usng these matr representatons to found the Bnet le formula for acobsthal numbers. A.J.Macfarlane [4] use the property for determnants and gve new denttes nvolvng Fbonacc and related numbers. Some determnantal denttes nvolvng Fbonacc polynomals, Lucas polynomals, Chebyshev Polynomals, Pell polynomals, Pell-Lucas polynomals, Veta-Lucas Polynomals are descrbed [5]. In ths paper, we ntroduce Generalzed Fbonacc-Le Polynomals and ts determnantal denttes. Also we establsh result n terms of Generalzed Pell Polynomals and Generalzed Pell- Lucas Polynomals.. GENERALIZED FIBONACCI-LIKE POLYNOMIALS We defne Generalzed Fbonacc-Le Polynomals by recurrence relaton, V ( ) V ( ) V ( ) ; n 3 wth V ( ) a, V ( ) b [.1] n n 1 n 1 Frst few polynomals are V ( ) b a 3 3 ( ) 4 ( ) ( ) ( ) ( ) V ( ) b a 4 V ( ) b 1 a 5 V ( 6 3 b If a b 1, then fn( ) fn 1( ) fn ( ) ; wth f1( ) 1, f( ) (Fbonacc polynomals) If a, b 1, then ln 1( ) ln ( ) ln 3( ) ; wth l0( ), l1( ) (Lucas polynomals) If a 1, b, then Pn( ) Pn 1( ) Pn ( ) ; wth P1( ) 1, P( ) (Pell polynomals) If a b, then Q ( ) Q ( ) Q ( ); wthq ( ), Q ( ) (Pell-Lucas polynomals) n 1 n n 3 0 1

3 Generalzed Fbonacc-le polynomal 1417 Now we defne a famly of Fbonacc-Le polynomal as { n ( ), n ( ), n m( ), n m ( ), n m ( )} V V V V V V, Where n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1. Then Generalzed Fbonacc-Le polynomals are V ( ) V ( ) V ( ) [.] n n n V ( ) V ( ) V ( ) [.3] n m n m n V ( ) V ( ) V ( ) [.4] n m n m n m If ( ab, ) (1,1), thenv ( ) f ( ), the Generalzed Fbonacc Polynomals. n n If ( ab, ) (,1), thenv ( ) l ( ), the Generalzed Lucas Polynomals. n n If ( ab, ) (1,), thenv ( ) P ( ), the Generalzed Pell Polynomals. n n If ( ab, ) (,), thenv ( ) Q ( ), the Generalzed Pell-Lucas Polynomals. n n If( ab,, ) (1,1,1), thenv (1) F, the Generalzed Fbonacc numbers. n n If ( ab,, ) (1,,1), thenv (1) L, the Generalzed Lucas numbers. n n If ( ab,, ) (1,1, ), thenv (1) P, the Generalzed Pell numbers. n n If( ab,, ) (1,,), thenv (1) Q, the Generalzed Pell-Lucas numbers. n n 3. DETERMINANTAL IDENTITIES Now we present determnantal denttes Theorem 1: If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then bvn ( ) avn ( ) V ( ) n V ( ) V ( ) av ( ) V ( ) bv ( ) bv ( ) 4 abv ( ) V ( ) V ( ) n n n n n n n bvn K( ) av ( ) av ( ) n n n n V ( ) b V ( ) n n av n ( ) Proof: Let Δ bvn ( ) avn ( ) V ( ) n V ( ) V ( ) av ( ) V ( ) bv ( ) bv ( ) n n n n bvn K ( ) av ( ) av ( ) n n n n V ( ) b V ( ) n n av n ( ) [3.1]

4 1418 V. K. Gupta, Y. K. Panwar and O. Shwal Assume bvn ( ) α, avn ( ) β, then by [.1] Vn ( ) α β, Now α β α β α β α β Δ α β ( α β ) α α β β α ( α β ) β Multplyng and dvded R1 by ( α β), R byα, R3 by β by α β ( α β) ( α β) 1 Δ α β ( α β) α αβ ( α β ) β β α ( α β) Applyng R1 R1 R R3 & R R1 R, R3 R1 R3 α β β ( α β) α ( α β) Δ β 0 ( α β) αβ ( α β ) α ( α β) 0 [3.] [3.3] [3.4] Epand along frst row, we get Δ 4 abvn ( ) Vn ( ) Vn ( ) [3.5] Put bvn ( ) α, avn ( ) β, Vn ( ) α β Theorem : If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then bvn () av n () Vn () bv n () Vn () Vn () b Vn () abvn () Vn () a V () bv () () { () () ()} n n Vn abvn Vn Vn abv () V () a V () av () V () V () n n n n n n Proof: Let bv () av () V () bv () V () V () n n n n n n n () n () n () () n n () n () abvn () Vn () av () av () () () n n Vn Vn Δ bv abv V av bv V [3.6] Assume bvn ( ) α, avn ( ) β, then by [1.1] Vn ( ) α β, Now α β( α β) α( α β) ( α β) Δ α αβ β α( α β) αβ β β ( α β ) ( α β ) [3.7]

5 Generalzed Fbonacc-le polynomal 1419 Tang α, β,( α β) common from C1, C, C 3 & Applyng R R ( R1 R3) α ( α β) α ( α β) Δ αβ( α β) 0 ( α β) ( α β) [3.8] β β ( α β) ( α β) Applyng C C C3& Epanson by R Δ ( αβ ( α β )) [3.9] Put bvn ( ) α, avn ( ) β, Vn ( ) α β, we get { abv ( ) ( ) ( )} n Vn Vn Δ [3.10] Theorem 3: If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then bvn ( ) abvn ( V ) n ( ) bvn ( V ) n ( ) abvn ( ) Vn ( ) a V ( ) av ( ) ( ) { ( ) ( ) ( )} n n Vn abvn Vn Vn [3.11] bv ( ) V ( ) av ( ) V ( ) V ( ) n n n n n Theorem 4: If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then a Vn () V () abv () () () () n n Vn bvn Vn abvn () Vn () bv () av () () { () () ()} n n Vn abvn Vn Vn [3.1] bv () V () av () V () a V () b V () n n n n n n Theorem 5: If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then Vn ( ) avn ( ) bvn ( ) bvn ( ) avn ( ) Vn () bvn () avn () Vn () avn () V () bv () av () bv () V () n n n n n { av } 3 n bvn Vn ( ) ( ) ( ) [3.13] Theorem 6: If n,,,, m, are postve ntegers wth 0 < <, 1 < m, 1, then 1 bvn ( ) avn ( ) 1 { abvn ( ) Vn ( ) Vn ( ) } 1 bvn () avn () Vn () V ( ) n { abvn () Vn () Vn () avn () Vn () bvn () Vn () abvn () Vn () } [3.14] Above Theorems 3 to 6 can be solved same as Theorem: 1.

6 140 V. K. Gupta, Y. K. Panwar and O. Shwal 4. CONCLUSION Ths paper descrbes Generalzed Fbonacc-Le polynomals and ts determnantal denttes. Also results derved n terms of classcal polynomals le Fbonacc Polynomal, Lucas Polynomals, Pell Polynomals and Pell-Lucas Polynomals. ACKNOWLEDGEMENT. The authors are grateful to the referees for ther useful comments. References [1] Aleandru Lupas, A Gude of Fbonacc and Lucas Polynomal, Octagon Mathematcs Magazne, Vol. 7, No.1 (1999), -1. [] A. Benamn, J. Qunn and N. Cameron, Fbonacc Determnants- A Combnatoral Approach, Fbonacc Quarterly, 45, No. 1 (007), [3] A. F. Horadam and Bro. J. M. Mahon, Pell and Pell-Lucas Polynomals, Fbonacc Quarterly, 3, No. 1 (1985), [4] A. J. Macfarlane, Use of Determnants to Present Identtes Involvng Fboncc and Related Numbers, Fbonacc Quarterly, 48, No. 1 (010), [5] B. Sngh, O. Shwal, Y.K.Panwar, Generalzed Determnantal Identtes Involvng Lucas Polynomals, Appled Mathematcal Scences, Vol. 3 (009), No. 8, [6] F. Koen and D. Bozurt, On the Jacobsthaal Numbers by Matr Methods, Int. J. Contemp. Math. Scences, 3(008), [7] N. Cahll and D. Narayan, Fbonacc and Lucas numbers s Trdgonal Matr Determnant, Fbonacc Quarterly, 4, (004), [8] Z. Spvey, Fbonacc Identtes va the Determnant sum property, College Mathematcs Journal, 37(006), Receved: January, 01