SOME PROPERTIES OF THIRD-ORDER RECURRENCE RELATIONS

Transcription

1 SOME PROPERTIES OF THIRD-ORDER RECURRENCE RELATIONS A. G. SHANNON* University of Papua New Guinea, Boroko, T. P. N. G. A. F. HORADAIVS University of New Engl, Armidale, Australia. INTRODUCTION In this paper, we set out to establish some r e s u l t s about t h i r d - o r d e r r e - c u r r e n c e relations, using a variety of techniques. Consider a t h i r d - o r d e r r e c u r r e n c e relation (.) S = PS, + QS. + RS Q (n 2s 4), S = n n - ^ n - 2 n - 3 u S where P, Q, R a r e a r b i t r a r y integers. Suppose we get the sequence (.2) { J ' } i when ^ =, S 2 =, S 3 = P s the sequence (.3) {K } 5 when S* «, S 2 =, S 3 = Q, the sequence (.4) { L }, when S t = f S 2 =, S3 = R It follows that Kt = J 2 - Ji, K 2 = J 3 - PJ 2 for n ^ 3, * P a r t of the substance of a thesis submitted in 968 to the University of New Engl for the degree of Bachelor of L e t t e r s. 35

2 36 SOME PROPERTIES OF [Feb. (.5) K = QJ - + RJ, n ^ n - n-2 ' (.6) L Q = B J ^. These sequences a r e generalizations of those discussed b y F e i n b e r g [ 2 ], [3] Waddill Sacks [ 6 ]. If the auxiliary equation 2. GENERAL TERMS x 3 - Px 2 - Qx - R = has three distinct real roots 3( suppose that they a r e given by #,j3,y. Accord: According to the general theory of r e c u r r e n c e relations, J can be r e p - resented by (2.) J n = Ac/ " + BJ3 " + Cy " where ~ ( 3 - a)(y - at> B ~ (y - P){a - fi) ' C = (a - y)(j3 - y) (A, B C a r e determined by J j, J 2, J 3. ) The first few t e r m s of { J } are L n J (Ji) =,, P, P 2 + Q, P 3 + 2PQ + R, P 4 + 3P 2 Q + 2PR + Q 2.

3 972] THIRD-ORDER RECURRENCE RELATIONS 37 These t e r m s can be determined by the use of the formula [n/3] [n/2] *» ' «. - E E ' E.) pn ~ 3 ~ 2) «' i= j= where a.. satisfies the partial difference equation (2.3) a.. = a a a i mj n - l, i, j n i? j - l n - 3, i - l, j with initial conditions a. noj F o r example, a. nio J 5 = a 3 P 3 + a 3 PQ + a 3 R T>3 = P d + 2PQ + R. F o r m u l a (2.2) can be proved by induction. In outline, the proof uses the basic r e c u r r e n c e relation (.) then the partial difference equation (2.3). The result follows because [(n-l)/3] [(n-l)/2]»!- E»' E VM,,^- 3 '- 2^ i= j= [(n-2)/3] [n/2] i= j=l

4 38 SOME PROPERTIES OF [Feb. [n/3] [(n-3)/2] n- Z ^ L^j n - 3, i - l, j i=l j= By using the techniques developed for second-order recurrence relations, it can be shown that (2.4) (P + Q + R - ) ^ J r = J n ( - P)J n ( - P - Q)J n + - l. r=l It can also be readily confirmed that the generating function for { j } is oo (2.5) 2 J n x I = x2( " P x ~ Q x2 " R x 3 ) ' n= 3. THE OPERATOR E We define an operator E, such that (3.) E J n = J n + l ' suppose, as before, that there exist 3 distinct real roots, a, fi 9 y of the auxiliary equation x 3 - Px 2 - Qx - R =. This can be written as (x - a)(x - j3)(x - y) = (x 2 - px + q)(x - y) =, where p ~ # + j 3 = P - y,

5 972] THIRD-ORDER RECURRENCE RELATIONS 39 q = aft. The r e c u r r e n c e relation J = P J + Q J + R J n n - ^ n - 2 n - 3 can then be expressed a s (E 3 - P E 2 - QE - R ) J = (replacing n by n + 3) o r (3.2) (E 2 - pe + q)(e - 7) J n =, which becomes (3.3) (E 2 - pe + q ) u n = o r n+2 ^ n+ ^ n if we let ( E - r ) j n = u n, where {u T is defined by (3.4) u n + 2 = p u n + - qu n, (n > ), u =, i^ = In other w o r d s, (3.5) u n = J n + - y j n the extensive p r o p e r t i e s developed for {u } can be utilized for { j }

6 4 SOME PROPEETIES OF [Feb. In particular? (3.6) u 2 - u. u,_, = q H n n - n+ becomes This gives us < 3 ' 7 > < J n + l " J n J n + 2 > + V n " Wn-> + ^K " W n - > = ^ Another identity for { j } analogous to (3.6) is developed below a s (4.4). Since J = u - + y J - r n n - n - = u - + y(u + J n ) n - fv n-2 n-2 = u i + y u + y 2 (u + J ) n - ' n-2 ' n - 3 n - 3 then (3.8) J n = X ^ " V l r = l n which may be a m o r e useful form of the general t e r m than those expressed in (2.) (2.2). 4. USE O F MATRICES Matrices can be used to develop some of the p r o p e r t i e s of these sequences. In general, we have

7 972] THIRD-ORDER RECURRENCE RELATIONS 4 P Q R' p Q R" 2 "Ssl i s 2 _ S J so, by finite induction; (4.) r s n s, n~l s n [_ n-2 = s 2 SiJ P Q R" fn-3 'S3] A gain j since p Q R j P 2 + Q PQ + R PR J 4 K4 RJ3 = P Q R J 5 K 3 RJ 2 J 2 K 2 RJi we can show by induction that p Q R" n J n+2 K n+2 R J n + l (4.2) s n = = J n + K n + RJ l J K RJ n - The corresponding determinants give (4.3) ( d e t S ) n = R n n+2 T n-f-l K. n+2 RJ n+ K n+ RJ. n J n K n RJ. n - By the repeated use of (.5), we can show that

8 42 SOME PROPERTIES OF [Feb. n+2 J n+ J n K 'n+2 K n+ K n RJ n+ RJ n RJ n - R 2 n+ F n+ J n n - n+ J n n-2 n - (4.4) n+2 T n+ n - n+ J n R n-2 J n n-2 T n- which is analogous to (4.5) u* - u - u,- n n - n+ n - for the second-order sequence {u } defined above, (3.4). In the more gene r a l c a s e, we get S = ~ n n+3 n+ n+2 n+2 n+ L n+ n - the corresponding determinants a r e S n+3 S n+ n+2 n+2 n n+ n - n+4 = R n - s 3 s A s 2 s 2 s S± Matrices can also be used to develop expressions for E ^n V ^Jl V ^ n= n! ' JLt n= n! ' Z-J n= n! 9

9 972] THIRD-ORDER RECURRENCE RELATIONS 43 by adapting extending a technique used by Barakat [] for the Lucas polynomials. Let a ll a 2 a 3 X a 2 a 22 a 23 a 3I a 32 a 33 with a trace P = a u + a 22 + a 33, det X = R, Q = y a., a.. - a., a.., (i ^ j) i,j=l F o r example, "a " X = / 3 y satisfies the conditions. The characteristic equation of X is A 3 - PA 2 - Q X - R = so, by the Cayley-Hamilton Theorem [ 4 ], X 3 = P X 2 + QX + RI Thus

10 44 SOME PROPERTIES OF [Feb. X 4 = P X 3 + QX 2 + R X = (P 2 + Q)X 2 + (PQ + R)X + P R j : so on, until (4.6) X n = J X 2 + K X + L I ~~ n ^ n -^ n ^ Now, the exponential of a m a t r i x X of o r d e r 3 is defined by the infinite s e r i e s (4.7) es = i + ^, x + A x +.. where I is the unit m a t r i x of o r d e r 3. Substitution of (4.6) into (4.7) yields (4.8) es = x ^ y j i + xy^ + iy ^ ^ * 4 n! ~ L^j n! ^L^j n! n= n= n= Sylvester s m a t r i x interpolation formula [5] gives us (4 " 9) 6 " 2 - f Ai 5 A 2, A3 X _ ^ Al (S - A 2 D(X - A 3 I) 6 (A t - A 2 )(Ai - A3) ' where Ai, A 2? A3 are the eigenvalues of X. Simplification of (4.9) yields E {e AjL (A 3 - A 2 )X 2 + e Xl (A - A )X + e Al A 2 A 3 (A 3 - A 2 > l } A,A 2,A3 (4.) e - = - (Ai - A 2 )(A 2 - A 3 ) (A 3 - Ai) By comparing coefficients of X in (4.8) (4.), we get

11 972] THIRD-ORDER RECURRENCE RELATIONS 45 Z e A i(a 3 -A 2 ) nt = n= Ai?xJ f A«^ " X * } \T^ Ln n= S e X l A 2 A3(A3 - A 2 ) TTcxt - A 2 ) The authors hope to develop many other properties of t h i r d - o r d e r r e c u r - rence relations. REFERENCES X. R. Barakat, "The Matrix Operator e the Lucas P o l y n o m i a l s, " Journal of Mathematics P h y s i c s, Vol. 4 3, 964, pp M. Feinberg, No. 3, 963, pp M Fibonacci - T r i b o n a c c i, " Fibonacci Quarterly, Vol., 3 e M. Feinberg, "New S l a n t s, " Fibonacci Quarterly, Vol. 2, 964, pp M. C. P e a s e, Methods of Matrix Algebra, New York, Academic P r e s s, 965, p H. W. Turnbull A. C. Aitken, An Introduction to the T h e o r y of C a n - onical M a t r i c e s, New York, Dover, 96, pp. 76, M. E. Waddill L. Sacks, "Another Generalized Fibonacci Sequence," Fibonacci Quarterly, Vol. 5, 967, pp