BASIC PROPERTIES OF A CERTAIN GENERALIZED SEQUENCE OF NUMBERS 1. INTRODUCTION. d = (p - 4q) ' p. a= (p +d)/2, P= (p - d)/2

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1 BASIC PROPERTIES OF A CERTAIN GENERALIZED SEQUENCE OF NUMBERS A. F. HORADAM The Uiversity of North Carolia, Chapel H i l l, N. C. Let a, (3 be the roots of 1. INTRODUCTION (1.1) 2 - p + q = 0 where p, q are arbitrary itegers. beig real, though this eed ot be so. Write Usually, we thik of a, p as (1. 2) d = (p - 4q) ' p The (1. 3) a= (p +d)/2, P= (p - d)/2 s o that (1. 4 ) a + P = p, ap = q, a - P = d. Recetly [6], a certai geeralized sequece w j was defied: (1.5) j w l = jw (a, b;p, q) : w Q = a, w ^ b, w = p w 4 - q w _ 2 ( ^ Z) i which (1.6) w = Aci + Bp, where (I 1) A - b " a P R - a a " b 1 1-7» A " a - p ii5 " a - P whece (1.8) A + B = a, A - B = (2b - pa)d -1, A B = e d" 2 i which we have writte 2 2 ( 1. 9 ) e = pab - qa - b. 161

2 162 BASIC PROPERTIES OF A CERTAIN Oct. Sequeces like jw [ have bee previously itroduced by, for eample, Bessel-Hage [ l ] ad Tagiuri [l l], though i the available literature I caot fid evidece of much p r o g r e s s from the defiitio [11] to have discovered a few of the results listed hereuder. The purpose of [6 J was to determie a r e c u r r e c e relatio for the k powers of w (k a iteger), that is, to obtai a eplicit form for w k () 2 w k _ =0 Here, we propose to eamie some of the fudametal arithmetical p r o p e r t i e s of jw 1. No attempt at all is made to aalyze cogruece or p r i m e umber features of 5w t. I selectig p r o p e r t i e s to geeralize we have bee guided by those p r o p e r t i e s of the related sequeces (see 2. below) which i the literature ad from eperiece seem most basic. Naturally, the list could be eteded as far as the r e a d e r ' s ethusiasm p e r s i s t s. It is iteded that this paper should be the first of a s e r i e s ivestigatig aspects of jw I. Orgaizatio of the m a t e r i a l is as follows: i 2», various special (kow) sequeces related to Iw [ a r e itroduced, while i 3. some liear formulas ivolvig jw I a r e e s - tablished, ad i 4. some o-liear epressios a r e obtaied. Fial- 2 iy, i 5., some commets o the degeerate case p = 4q a r e offered,, 2. RELATED SEQUENCES j f j, P a r t i c u l a r cases of 5w I a r e the sequeces ^u I, 5v t, jh I, jl [ give by: (2.1) w (1, p; p, q) = u (p, q) (2.2) w (2, p; p, q) = V (p, q) (2.3) w (r, r + s ; 1, -1) = h (r, s)

3 1965 GENERALISED SEQUENCE OF NUMBERS 163 (2.4) w ( 1 ; 1; 1, -1)= f ( = u ( l, - l ) = h (l,0)) (2.5) w (2, 1; 1, -1)= l ( = v ( l, - l ) = h (Z,-l)). Historical iformatio about these secod order recurrece sequeces may be foud i Dickso [3]. Of course, Jf I is the famous Fiboacci sequece, il I is the Lucas sequece, ad Ju [ ad Jv I are geeralizatios of these, while Jh 1 discussed i [4] is a differet geeralizatio of them. Chief properties of ju I, Jv [, Jf [ ad il [ may be foud i, for istace, Jarde [7], Lucas [81 ad Tagiuri [10] ad [l l], those of Jf I especially beig featured i Subba Rao [9j ad Vorob'ev [l 2]. Two rather iterestig specializatios of (2. 1) ad (2. 2) are the Fermat sequeces Ju (3, 2)1 =J2 -ll ad Jv (3, 2)1 = j2 + l[, ad the Pell sequeces Ju (2, -1)[ ad Jv (2, -1)1. (See [l] or [8]).. From (1.6), (1.7) ad (2.1) - (2.5) it follows that +1 fl+l (2.6) u = ^ ' d (2. 7) v = a + p (2.8) h _ (r + s - r P ^ a " - (r r a ^ p " (2-9) f _ a l " h ^5 (2.10) 1 = a? + p! 1 v ' i l wherei l + N / /9 \ ^ (2.11) a o 1 -^5 = -,, p = that is, a,, (3 are the roots of (2.12) = 0.

4 164 BASIC PROPERTIES OF A CERTAIN Oct. Cosequetly, by (1.4) (2. 13) a + P = 1, a 1 P 1 = - 1, c^ - ^ = 5. To a s s i s t the r e a d e r, ad as a source of ready referece, the full set of results for the five specializatios of Jw { will ofte be writte dow, as i (Z. 6) - (Z. 10). Obviously from (1. 9), e c h a r a c t e r i z e s the various sequeces. F r j U j > 2 j V ( ' j h ' j f [ ' \* \ r - r s - s, 1, 5 respectively. By (1. 6), (1. 7) ad (Z. 6) we have W e d e r i v e e = ^ ^ " ^' (Z 0 14) w = au + (b - pa) u, = bu, - q a u 0, v v r ' ' -1-1 ^ -Z with, i particular, the kow [8] epressios (Z. 15) v = Zu - pu, = pu, - 2q u -. c r ' -1-1 ^ -Z (Ultimately, 7 of course, these yield 7 1 = Zf - f. + Zf ~. ) -1 -Z Puttig = 0 i(z. 14) r e q u i r e s the eistece of values for egative subscripts, as yet ot defied. Allowig u r e s t r i c t e d values of therefore i (1. 6) we obtai (2.16) N after simplificatio usig C w' = A a" + B (3~ = q Ki^V-l* (Z. 17) u = q u ^, ' - ^ -Z which follows from (Z. 6). Combiig (Z. 14) ad (Z. 16) we have (au - bu -,) /o i o\ - ' - 1 ' (Z.18) w = q r-, w v H ' - au + (b - pa u T r -1 whece it follows from (Z. Z) - (Z. 5) that (2; 19) v = q v ' - ^

5 1965 G E N E R A L I S E D S E Q U E N C E O F N U M B E R S 165 \r (u - u, ) - su A ( ) h = ( - l ) «- 2 l l 2 l i? h - r u -f s u, - 1 < > ^ ^ " 1 ^ I p a r t i c u l a r, ( ) w, = A a" 1 + (3" 1 = P a " b -1 q so t h a t (2. 24) u l - 0 (2. 25) - 1 V = p (2..26) h - l - s ( ) f = 0 (2. 28) l = -1 M a y of the s i m p l e s t Iw \ a r e e p r e s s i b l e i t e r m s of jf 1 B e s i d e s ( 2. 4 ) we h a v e ( ) w ( _ 1 ' 1 ; - 1 ' _ 1 ) = < " 1 ) " 1 f ( ) w ( 1, - 1 ; 1, -1) = -f _ 3 ( ) w ( 1, 1; - 1, -1) = ( - l ) _ 1 f 3. M o r e g e e r a l l y, ( ) w (a, b; 1, -1) = af _ 2 + b f ^ ( ) w (a, b; - 1, -1) = ( - l ) k f 9 - bf A y I -Z -1) N o t i c e t h a t (2. 34)<[provided w ( a l f b i ; p r q ) = - w ( a 2, b 2 ; p 2 > q 2 ) a 2 = ' a r b 2 = " b i ' P 2 = p r q 2 = q r

6 166 BASIC PROPERTIES OF A CERTAIN Oct. Some sequeces a r e cyclic. Eamples a r e (2.35) w (a, b; - 1, 1) 2 for which a, (3 (= a ) a r e the comple cube roots of 1 ad (2.36) w (a, b; 1, 1) 2 for which a, (3 (= a ) a r e the comple cube roots of - 1. Sequece (2. 35) is cyclic of order 3 (with t e r m s a, b, -a - b) sice a = p = 1, while sequece (2.36) is cyclic of order 6 (with t e r m s a, b, -a + b,, u\ 3 03, 6 Q6,,,,. ^. -a, -b, a - b) sice a = p = - 1, so a = p = l ( odd m this case). (Refer (1.6)). Geometric-type sequeces a r i s e whe p.= 0 (so that by (1.5) w + 1 = - q w 4 ) ad q = 0 (so that w + 1 = p w j. 3. LINEAR PROPERTIES F r o m (1.5) ad (1.6) it follows that (3.1) w w, -1 CL P.. k w (a w - k. > (/P if if -1 < p < 1, -1 < a < 1, 2 (3. 2) w (p - q) w + pq w ^ = 0, ad (3. 3) P W (p 2 - q) w q 2 w _ = 0. Repeated use of qw,, = -w,,, + pw (k = 1,..., ) leads to the sum of the first t e r m s -1 (3. 4) q 2 ] w^ = (p -1) (w 2 + w w ) - w p W]L whece -1 (3.5) (p - q - 1) ^ w = w w -(p - 1) (w - w Q )

7 1965 GENERALISED SEQUENCE OF NUMBERS 167 while the correspodig r e s u l t s for differeces are -1 (3.6) q ] T (-1) J w. = (p +1) (-w 2 + w 3 + ( - 1 ) _ 1 W ) + ( - 1 ) I l w + l + p W l ad - X (P - q + D X (-1) w. (3.7) j " = ( - l ) + 1 w + 1 +w 1 -(p+1) j ( - l ) + 1 w + w Q [. Replace by 2 i (3. 4), (3. 5) (3. 6) ad (3. 7). Write (3.8) < 7 = w 0 + w w 2 _ 2, ad (3.9) P = w + w w 2 _ 1. Addig ad subtractig (3. 4), (3. 6) give (3.10) (1 +q) (J = - p p - ( w 2 - w Q ) ad (3.11) (1 +q) p = p O +q(w 2 _ 1 - w_ ) for the sum of the eve - (odd -) ideed t e r m s of ^w \. Clearly by (1.5) additio of (3. 10) ad (3. 11) yields the sum of the first 2 t e r m s (3. 4) as epected. Solve (3. 10) ad (3. 11) so that (3.12) j p 2 - (l+q) 2 [ a = (l+q)(w 2 - w Q ) - pq ( w ^ ^ - w ^ ) ad (3.13) j p 2 - (l+q) 2 [ 9 = p ( w 2 - w Q ) - q(l+q)(w 2 _ 1 - w ^ ). Usig the alterative epressio w = bu, - qau ~ (2. 14), we have ( +1 I ^ 0-1 \ / w +^ i o = w~ 2 u - q w, 1 u -1, f w. 0 = w 0 u V q w~ u, 2-1

8 1^8 BASIC PROPERTIES OF A CERTAIN Oct. whece C \ w +r, = w r u - q ^ w r - 1, u -1, (3.14) < i = w u - q w 1 u 1 I r ^ -1 r - 1 o iterchagig ad r. Equatios (3.14) may also be obtaied from (1.5), (2. 1) ad ( ). (3.15) Of course w, = w. u,. - q w.. u,., +r r - j +j ^ r - j - 1 +j-1 +j r - j ^ +j-1 r - j - 1 also. Further, from (1. 6) ad (2. 7) it follows that /o i \ w. + r, r + q w (3.16) = v w r t h a t i s, the epressio o the left is idepedet of a, b,. r ad i (3. 16) ad the set r = 0. Accordigly, -r Iterchage (3. 17) w + q w = a v. ^. - Observe also from (1. 6) ad (2. 6) that r w. - q w u, (3.18) - ± I " r r _ 1 s u, 7. - q w s -1 which [lo] is a iteger +s provided ^ s - s divides r. Two biomial results of iterest may be oted. Firstly, from (1.6) it follows that (3.19) w 2 = ( " q ) Z ( j> ( - > w --< where we have used the fact a^ - pa + q = 0, (3^ - pp + q = 0. Startig from (1. 3) ad (1. 6), we readily derive 2 w = A(p + d ) + B(p - d )

9 GENERALISED SEQUENCE OF NUMBERS l6<? [/2] (3.20) 2 w = a Z p _ 2 j d Z J ( J ) [-lj j=o 2 J + ( 2 b - p a ) Z J=0 ( * J J p ^ ^ d 2 ; whece follow the kow [ l ] epressios (3.21) 2 u = [ ] ( J ) p - 2 J d 2 J (3.22) 2 " 1 v = ^ ( J ) p - 2 J d 2 J [ / 2 ] +1 i (3.23) 2 f = <2j+V 5 (3.24) for 2 _ l l = /2] Z ( 2j ) ^ * Suitable substitutios i the above results lead to the special c a s e s K l * j V ' J M ' j f j ad j l j ; for eample, for j f j, i (3.4) ad i (3. 14) with r =, < T + P = f 2 + l - 1 ' usig (3. 19). If we write + f - l = f 2 = <& f - k ' k=0 (3.25) % r W +1 so that, by (1.5), l (3.26) r =, r - p - q r ' -1 - p - q r 2

10 170 BASIC PROPERTIES OF A CERTAIN Oct. eablig us t o e p r e s s the limit of the ratio as a cotiued fractio. Sometimes, whe q = - 1, it is otatioally coveiet to write where (1.2) a - e " = smh f] + cosh Q O 'O - 7] - sih T) - cosh p = -e o 'o b o - o -1 (3. 28) cosh T) = ', sih r\ =, tah t) - p d * ' o 2 ' 'o 2 o ^ o Zero suffices sigify that q = - 1. Combiig this hyperbolic otatio with the r e m a r k s immediately precedig (3.27), ad proceedig to the limit (refer (3.1)), wevsee that for p - 1, q = - 1, that is, for \h I (ad its specializatios Si \, Si (),. 1 - r \\ T- r - e h i.. l a "1 cosh 7 - sih r\ (observe that by (2.12) = g is a root of = 0 so t h a t i ttl g - } leadig to the cotiued fractio. ) F u r t h e r m o r e, (3.27) ad (3.28), with (1.5), imply (3130) w =(A + (~l) B ) sih rj + (A - (-1 ) B ) cosh ri o, o ' o 'o o % ' o' 'o Hyperbolic epressios for the specialized sequeces a r e the, (2.6), (2.7), (2.9), (2.10), from (3.31) J sih ( + 1) 77 u = T ( odd) cosh ' cosh ( + 1) 71, cosh 'o ( eve) o

11 1965 G E N E R A L I S E D S E Q U E N C E O F N U M B E R S 171 ( ) ' v = 2 s i h T) ( eve) 2 c o s h r) ( odd) w i t h c o r r e s p o d i g e p r e s s i o s for f, 1 r e s p e c t i v e l y, i w h i c h r\ i s r e p l a c e d by T]. A h y p e r b o l i c e p r e s s i o for h i s g i v e i [5]. E s s e t i a l l y, 4. N O N - L I N E A R P R O P E R T I E S the p r o b l e m i o b t a i i g o - l i e a r f o r m u l a s (as i the l i e a r c a s e ) is to d e t e c t the a p p r o p r i a t e c o e f f i c i e t s (fuctios of k p, q) of w. B a s i c o - l i e a r ( q u a d r a t i c ) r e s u l t s h a v e a l r e a d y b e e r e c o r d e d i [6 J, a m e l y ; (4. 1) a w, + ( b - rp a ) w,. - w w - qw, w,, ' m + m m ^ m (4. v 2) a w 0 + v ( b - rp a ) w~, = w - qw, = w., w, - q w w 0, ' 2 ' 2 - l ^ ^ - 2 ia o\ 2-1 (4. 3) w., w, - w = q e. ' +1-1 O b v i o u s l y, f r o m (4. 3) w i t h = 0, (4, 4) e = q (w l w ^ - w Q ) w h i c h m a y be c o m p a r e d w i t h ( 1. 9), u s i g ( 1. 5) ad (2. 23). A e t e s i o of (4. 3) i s, by ( 1. 6) ad (2. 6), / A r\ 2 - r 2 (4. 5) w, w - w = e q u,. v ' + r - r r - 1 P u t t i g r = i ( 4. 5 ), we h a v e 2 2 (4. 6) w + e u T = a w 0 ' I t e r c h a g e r ad i ( 4. 5 ), t h e s u p p o s e r =. 0. We d e d u c e / A ~7\ (4. 7) w w = a + e q u,. ' - ^ - 1 ( = 1 r e d u c e s (4. 7) to (4. 4). ) S p e c i a l i z a t i o s of (4. 1) a r e, o m u l t i p l i c a t i o by 2 ad u s e of ( 1. 2 ), ( 1. 4 ), ( 2. 6 ), ( 2. 7 ) a d (2. 15), t h e k o w [8] r e s u l t s

12 172 BASIC PROPERTIES OF A CERTAIN Oct. (4. 8) 2 u, 1 = u. 1 v + u, v m+-1 m m ad (4. 9) 2 v.. = v v. + d u 1 u 1. ' m+ m m Net, by (4.6), we derive, usig (2. 6), (2.7), (1. 2) ad (1. 4), (4.10) u-., = u, v ' 2-l -1 ad (4. 11) 2 v, = v 2 + d 2 u with i A i o\ 2 0, 2 2 I 0 (4. 12) v. = v - 2q = d u, + 2q. 2-1 ^ Agai, (4.1) with m = 2 gives a epressio for w~ from which we deduce, by (4. 10), (2.6), (2.7) ad the r e c u r r e c e relatio for v~, 3 /A i o\ 3-l 2 (4. 13) = v - q ad u T -1 V-2 0 / A i A \ 3 L 0 (4. 14) = v - 3q ' v Results (4. 10) - (4. 14) occur i Lucas [8] i a slightly adjusted otatio. Comig ow to the sum of the first half of (4. 2). Write -1 (4.15) ^ = Z w 2 j The, it follows that 2 (4.16) (1-q) r = a ( 7 + ( b - p a ) p - j ^ ( b " P a ) w 2 - l 1 H t e r m s, we use the first

13 1965 GENERALISED SEQUENCE OF NUMBERS 173 whece r may be foud from (3. 12) ad (3. 13). Repeatig the first half of (4. 2) leads to (4.17) w ^ - q 2 w ^ = b w <b- p a ) q w 2-1. From (1. 6), (1.8) ad (2.6), (4. 18) w w - w w.. = q e u, u.,, -r +r+t +t ^ r-1 r+t-1 whece t = 0 gives (4.5) Repiac Replacig w by u i (3. 14) ad (3. 15) (with -j substituted for j) yields. - u u - q u, u T = U. u,. - q u i u.., +r r -1 r-1 -j r+j ^ -j-1 r+j-1 whece u u - u. u,. = q (u, u, - u., u..,) r -j r+j ^ -1 r-1 -j-1 r+j-1 (4.20) { -j,. ' ^ - q J (u. u... - u, 9.) ^ j r-+j r-+2j -j+1 - q J u. -, u,. - H j-1 r-+j-1 by repeated applicatio of (4. 19) ad replacemet i the first half of (4. 19) of by r-+j ad r by j to obtai a epressio for u,,. (u = 1). Note that (4.20) is the special case of (4.18) for which w = u so that e = -q (, r, j i (4. 20) replaced by - r, + r + t, ^ respectively ad (2.17) used). I particular, it follows from (4.20) with j = 1 that (4. 21) u, u -> - u ^ u, ~ q u,. v ' -1 r-2-2 r-1 ^ r--1 Moreover, (4.21) ad w = b u -, - qau 0 give for the se ' & queces (w \ ad jw' ( (4.22) w' w - w w' = q (a* b - a b! )(u u ~ - u *, u-, ) v ' r r ' -1 r-2-2 r-1 = q (a'b - a b') u, ^ ' r--1 Cubic epressios i w are geerally quite complicated, so we derive oly the sum of the first cubes, Cube both sides of(1.5)ad the use (1.5) agai. Thus

14 174 BASIC PROPERTIES OF A CERTAIN Oct (4. 23) w,, = p w - q w, - 3 pq w * w w,. r ' +1-1 ^ But, from (4. 3), (4. 24) w, w w,, = w + q e w, v ' ^ so that from (4. 23) ad (4. 24) it follows that O O Q. *2 ' "2 (4. 25) w (3 pq - p ) w + q w ^ = -3 pq 11 e w. Now a calculatio ivolvig (1.6) ad the summatio of geometric s e r i e s leads to -1. ^ 1 o (4.26) q J w = 3 j w - q w 0 - q ' ( w - q w ^ ) \. j = 1 i-pq+q Write - 1 (4.27) co = X w 3 J Combiig (4. 25), (4. 26) ad (4. 27), we fid (4,28) (l+3pq-p 3 +q 3 )a; = ^ E S _ j w ^ q ^ - q * 1 " V q K ^ \ 1-pq+q q w _ r w + (l+3pq-p ) w Q Appropriate substitutio i the above formulas of 4. Lead to correspodig results for the special sequeces (2.1) - (2.5). For istace, applyig (4. 16) ad (4. 28) to if I, we have r ^ _ respectively. co = I J'f 3 + f 3 + St-l) 11 "* 1 f, + z\ 4 < -1 ' -2 > 5. DEGENERATE CASE i f 2-rVi! Throughout the aalysis of the ature of Jw [, the hypothesis 2 2 that p $ 4 q has bee a s s u m e d. But suppose ow that p = 4 C 1. The

15 1965 GENERALISED SEQUENCE OF NUMBERS 175 simplest degeerate case occurs whe p = 2, q = 1 (a = p = 1) for which eists the trivial sequece ( ^ 0) (5.1) v (2, 1) : 2, 2, 2, 2, 2,.... ad the sequece of atural umbers ( > 0) (5.2) u (2, 1) : r, 2, 3, 4, 5,..., that is, u = +1 ad v = 2. For egative to, (2. 19) implies r v =v. ~ that is, every elemet of ju (2, 1) is 2, while (2. 17) implies u = -u 2, that is, like elemets of Ju (2, 1)1 a r e the positive ad egative itegers i o r d e r. Geerally, i the degeerate case, ( 5. 3 ) a = p =. The mai features of the degeerate case, as they apply to ju ad 5v a r e discussed i Carlitz [2], with ackowledgemet to Riorda. Brief commets, as they relate to Sw I, a r e made i [6 J. I passig, we ote that Carlitz [2] has established the iterestig r e - latioship betwee degeerate K < P. > ( ad the Euleria polyomial A, () which satisfies the differetial equatio A,, () = (1 + ) A () + (l - ) ^ - A (), +1 d 2 where A () = A, () = 1, A~() = 1+, A A) = Fially, it must be emphasized that jh [ ad its specializatios jf ad jl ca have o such degeerate c a s e s, because p - 4q the equals 5 ( + 0). I REFERENCES 1. Bessel-Hage, E., "Repertorium der hbhere Mathematic, " Leipzig, 1929, p

16 176 BASIC PROPERTIES OF A CERTAIN Oct. 2. Carlitz, L., "Geeratig fuctios for powers of certai s e - queces of u m b e r s, " Duke Math. J., Vol. 29, 1962, pp Dickso, L,, "History of the theory of u m b e r s, " Vol. 1, New York, 1952, Chapter Horadam, A. F», "A geeralized Fiboacci sequece, " A m e r. Math. Moth., Vol. 68, No. 5, 1961, pp Horadam, A. F., "Fiboacci umber t r i p l e s, " A m e r. Math. Mothly, Vol. 68, No. 8, 1961, pp Horadam, A. F., "Geeratig fuctios for powers of a certai geeralized sequece of u m b e r s, " (to appear i Duke Math. J. ). 7. Jarde, D., "Recurrig s e q u e c e s, " Riveo Lematematika, J e r u s a l e m, Lucas, E., "Theorie des o m b r e s, " P a r i s, 1961, Chapter Subba Rao, K., "Some p r o p e r t i e s of Fiboacci u m b e r s, " A m e r. Math. Moth., Vol. 60, No. 10, 1953, pp Tagiuri, A., "Recurrece sequeces of positive itegral t e r m s, " (Italia), Period, di Mat., s e r i e 2, Vol. 3, 1901, pp Tagiuri, A., "Sequeces of positive i t e g e r s, " (Italia), ibid. pp Vorob'ev, N» N», "The Fiboacci u m b e r s, " (traslated from the Russia), New York, A. F. Horadam Uiversity of New Eglad, Armidale, Australia. XXXXXXXXXXXXXX REQUEST The Fiboacci Bibliographical R e s e a r c h Ceter d e s i r e s that ay r e a d e r fidig a Fiboacci referece, sed a card givig the referece ad a brief descriptio of the cotets. P lease forward all such iformatio to: Fiboacci Bibliographical R e s e a r c h Ceter, Mathematics Departmet, Sa Jose State College, Sa Jose, Califoria