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A Generalization of Recursive Integer Sequences of Order 2 Stephen A. Parry Missouri State REU August 1, 2007 Synopsis A ring extension from Ω to Ωα. The definition of Period-1 and Period-2 orbits, along with an algorithm for finding Period-2 matrices. Discuss isomorphisms Generalized Relations Powers of Period-1 and Period-2 2x2 Matrices Multiplying NxM Recursive Matrices Generalizations Problems that need additional research Definitions Definition: A recursive integer sequence of order 2 is a sequence in the form An An1 An2 , , Z. Definition: The Fibonacci sequence is a recursive sequence Fn Fn 1 Fn 2, F 1 1, F 2 1. {...1,1, 2,3,5,8,...} Definition: The Lucas sequence is a recursive sequence Ln Ln 1 Ln 2, L1 1, L 2 3. {...1,3, 4, 7,11,18,...} Definition: a b b M (2, Z) b a Recursive Matrix We will use this definition of a recursive matrix interchangeably with the recursive sequence representation. Definition: A recursive matrix is an nxm matrix in the form An A n 1 An r An 1 An c An r c Note: The recursive matrix need not be square. Recursive Matrices Example: An 2 A n 1 4 A n 2 A1 1, A2 2 {...1, 2,8, 24,...} 24 8 8 2 Example: {...1, 2,8, 24,...} 24 8 8 2 2 1 An A n 1 A n 2 A1 1, A2 1 {...1,1, 2,3,5,8,...} 2 1 1 1 {...1,1, 2,3,5,8,13,...} 13 8 5 3 8 5 3 2 5 3 2 1 3 2 1 1 More Definitions Definition: F is the set of recursive matrices with integer multiples of Fibonacci entries. Definition: L is the set of recursive matrices with integer multiples of Fibonacci entries. Examples: 2 3 1 F 1 18 11 7 7 4 3 4 3 1 4 1 2 L 2 1 1 1 F, About Ω Theorem: Ω forms an integral domain. Definition: We define σ as the shift map. Theorem: Theorem: F n 2 F1 a n F1 1 1 F0 n 1 F nZ 0 4 3 A L 3 1 F4 k F A2 k 5k 4 k 1 F F F 4 k 1 4k L4 k 2 L A2 k 1 5k 4 k 3 L L4 k 2 L4 k 1 Examples Fibonacci * Fibonacci 2 1 3 2 1 1 2 1 8 5 F 5 3 Fibonacci * Lucas 2 1 4 3 1 1 3 1 11 7 L 7 4 Lucas * Lucas 7 4 4 3 4 3 3 1 8 5 5 F 5 3 l L f f F f F F L F L Ring Extension Now we have a result for the Fibonacci and Lucas numbers, we would like to make a generalization for all recursive sequences, An An1 An2 , , Z. The matrix we would require in the 2x2 case is b a b b . a Since the identity is in Ω, then we must have the identity in the ring extension. Thus, β=1. Therefore, we will concentrate on An An1 An2 , Z. Ring Extension b a b a b b a b b a More Definitions Definition: Definition: Pw1 A Period-1 sequence is any sequence when expressed in matrix form will be closed under multiplication. Define this set as Pw1 . A Period-2 sequence is any sequence when expressed in matrix form will be closed under multiplication in the union of the Period-2 sequence and its complimentary Period-1 sequence. Define this set as Pw2 . Pw2 R Cw Isomorphism Theorem: Theorem: Z[ ] Z[ ], 1 1 0 (1 )2 n / 2 1 A2 n 1 A2 n such that An An 1 An 2 , A1 1, A2 . Example: (1 )3 / 2 1 (1 3 3 2 3 ) / 2 1 5 8 F5 F6 Period-2 2x2 Matrices Theorem: We express a Period-1 sequence as {...1, , 2 1, 3 2 ...}. 2 1 . 1 Theorem: We express a Period-2 sequence for odd α as {... , 2 2, 3 3 ...}. 3 3 2 2 2 . 2 Theorem: We express a Period-2 sequence for even α as 1 {... , 2 2, 3 3 ...}. 2 1 3 3 2 2 2 2 2 Proof for odd α Definition: Given a recursive sequence in the form, An An1 An2 we define the characteristic polynomial, x Proof: 2 4 2 2 4 2 x 2 x 1. . . 2 1 2 4 2 2 4( 1) ( 2 2) Note: The discriminant of the characteristic polynomial plays an important role. Proof for odd α 3 3 2 2 2 2 6 7 4 13 2 4 5 6 3 8 5 3 4 2 6 8 5 4 2 4 2 3 3 1 2 2 2 2 ( 4) ( 4) 3 2 1 2 2 n n Pw1 2 n 1 Pw2 α=5 Example: Period-1 Period-2 {...1,5, 26,...}. {...5, 27,140...}. 26 5 5 . 5 1 140 27 5 . 27 5 26 5 5 1 5 1 1 0 135 26 26 5 140 27 27 5 27 5 5 2 135 26 29 26 5 26 5 5 1 5 1 5 1 1 0 1 0 701 135 135 26 140 27 27 5 27 5 27 5 5 2 5 2 3775 727 29 727 140 Generalized Recursive Relations A2 Definition: A1 Theorem: Theorem: A1 A0 is the shift map Z.. Fn 1 Fn F F n 1 n An 1 An n A n An 1 n 1 Theorem: An m An1 Am An Am1 Proof: Theorem: Proof: m n 1 m n1. A22k 1 A2 k A2 k 1 A22k 1 det(n ) 1. The Shift Map Example: 2. 2 1 2 1 0 Period-1 {...0,1, 2,5,12...} Period-2 {...0,1,3, 7,17...} 7 3 3 1 5 2 2 1 5 2 2 1 12 5 2 1 1 0 5 2 7 3 2 1 17 7 3 1 1 0 7 3 {...0,1, 2,5,12...} {...0,1,3, 7,17...} n n Pw1,2 Pw2,2 A Characteristic of Period-1 Matrices Period-1 Matrices act as units to their correlating Period-2 Matrices n Theorem: Let E , then E Z Proof: b a b B3 B b a 2 B2 B1 2b a b b a B4 b B3 b a B n n 3 Bn 2 Bn 2 . Bn 1 n E B3 B2 Powers of Λα and ηα Theorem: Theorem: A2 n1 A2 n n For α even A2 n A2 n1 2 n 1 1 4 2 n 1 2 n 2 n 2 n Theorem: For α odd 2 2n 1 4 2 n 1 4 2 n 1 2 n2 4 2 n 2n 2 n Periodicity 2 n n n Theorem: If C , then C , or C b a b Proof: C a b ( b a ) 2 b 2 2 C b 2ab X ( b a ) 2 b 2 Y b 2ab Z b2 a 2 X 0, Z 0, let b 2ab 0 b( b 2a ) 0 2a b 0, b b 2ab b2 a 2 If b 0, 1 0 C a n 0 1 2a If b , 2 2 2 3 C a 2 2 1 a k , odd ak 2 , even 4x4 Period-1 and Period-2 Matrices Example: Note: Similar to Period-1 and Period-2 2x2 matrices, it is possible to create a general formula for every α. Higher Degree Periods Theorem: For n 1, 2 a, b, c Z such that the primitive case of (a b )n c n is true. Proof: The proof is dependent on the fact the degree of the ring extension is 2. Q= ( ) n c Ωα Ω 2 Q= nc Thus, there fails to exist periods of degree greater than 2. The NxM case Definition: A complete orbit is an orbit closed under Pw1 Pw2 . 11 7 C1 7 4 L 4 3 Example: 34 21 13 C1C1T 5 21 13 8 F 13 8 5 573 361 C1C1T C1 5 354 223 F, L 219 138 F L This complete one orbit, not two; this fails to be a complete orbit. Resulting Generalizations Theorem: We are guaranteed a complete orbit when we are given nxm matrix where n Z., m 2Z.. Theorem: Every nxn recursive matrix, Mn, where n is even , forms a ring. Theorem: The set of nxm matrices that form a complete orbit is a semigroup. Theorem: If C 2 P1 , then C P1 , or C P2 Pw1 Pw2 Problems of Interest Relations between Period-1 and Period-2 Sequences: ( Bn Bn 1 ) 2 k k ( A1 A2 ) k ( n 1) . Finding more isomorphisms Continued fraction maps Eigenvalue maps Determinant maps Forming relationships for any power of every nxm recursive matrix in the Period-1 and Period-2 sets. Studying recursive relations of greater order. References [1] George E. Andrews. Number Theory. Dover, 1994. [2] Lin Dazheng. Fibonacci matrices. Fibonacci Quarterly, 37(1):1420, 1999. [3] Michele Elia. A note on fibonacci matrices of even degree. International Journal of Mathematics and Mathematical Sciences, 27(8):457-469, 2001. [4] John B. Fraleigh. A First Course in Abstract Algebra. AddisonWesley, seventh edition edition, 2003. [5] Ross Honsberger. Mathematical Gems III. MAA, 1985. [6] T.F. Mulcrone. Semigroup examples in introductory modern algebra. The American Mathematical Monthly, 69(4):296{301, Apri., 1962. [7] N.N. Vorobyov. The Fibonacci Numbers. The University of Chicago, 1966. [8] Lawrence C. Washington. Some remarks on fibonacci matrices. Fibonacci Quarterly, 37(4):333-341, 1999. [9] Kung-Wei Yang. Fibonacci with a golden ring. Mathematics Magazine, 70(2):131-135, Apr.,1997.