On the Product Representation of Number Sequences, with Applications to the Family of Generalized Fibonacci Numbers

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1 Joura of Iteger Sequeces, Vo , Artice O the Product Represetatio of Number Sequeces, with Appicatios to the Famiy of Geeraized Fiboacci Numbers Michee Rudoph-Liith Uité de Neuroscieces, Iformatio et Compexité UNIC CNRS, 1 Ave de a Terrasse Gif-sur-Yvette Frace rudoph@uic.crs-gif.fr Abstract We ivestigate geera properties of umber sequeces which aow expicit represetatio i terms of products. We fid that such sequeces form whoe famiies of umber sequeces sharig simiar recursive idetities. Appyig the proposed idetities to power sequeces ad the sequece of Pochhammer umbers, we recover ad geeraize kow recursive reatios. Restrictig to the cosie of fractioa ages, we the study the specia case of the famiy of k-geeraized Fiboacci umbers, ad preset geera recursios ad idetities which ik these sequeces. 1 Itroductio It has og bee kow that Fiboacci A ad Pe A umbers, defied by F 0 = 0,F 1 = 1,F = F 1 +F 2 1 ad P 0 = 0,P 1 = 1,P = 2P 1 +P 2 2 1

2 with 2, respectivey, ca be represeted i product form see, e.g., [1, 2, 3, 4, 5], specificay π π F = 3+2cos = 1 2icos 3 ad P = cos 2π = 1 2 2icos [ ] π, 4 where N, 2. The above sequeces F ad P are specific exampes of geera Lucas sequeces, the atter beig defied by the recursive reatio L m,p 0 = 0,L m,p 1 = 1,L m,p = ml m,p 1 pl m,p 2 5 for 2 [4]. Aready Zeiti [3] showed that L m,p = p m p 2cos π, 6 m, p R, provides a vaid product represetatio of a members i geera Lucas sequeces. Later, expressios of the form 6 were used to obtai other expicit represetatios of the correspodig umber sequeces i terms of fiite power series i the sequece parameters see, e.g., [6, 7, 8, 9], thus highightig the importace ad usefuess of such product represetatios for the ivestigatio of umber sequeces. I this cotributio, we wi show that product represetatios of umber sequeces ca aso be utiized to estabish direct iks betwee differet sequeces. To that ed, we formuate Defiitio 1. famiy of umber sequeces Let {x, } with x, C ad, N be a arbitrary two-parameter set of umbers. The correspodig famiy of umber sequeces {X,m } is defied by the set of a X,m with X,m = m+x,, 7 where m C abes the idividua umber sequeces withi the famiy, ad the members of each sequece. I what foows, we wi restrict to the subset of famiies which are costructed by m Z. A cocrete exampe of such a famiy is give if we set x, = 2icos π +1. I this case, usig 6 with p = 1 ad m Z, we have X,m = m 2icos π m, 1 +1 = L +1, 8 2

3 thus X,m defies the famiy of geeraized Fiboacci sequeces F m obeyig the recursive reatio := L m, 1, 2 F m 0 = 0,F m 1 = 1,F m = mf m 1 +F 2. m 9 We wi ca this famiy the Fiboacci famiy, ad expore some of its properties i Sectio 4. The paper is orgaized as foows. I Sectio 2, we wi prove various geera properties of the umber sequeces X,m withi a give famiy, with focus o iear recursive reatios betwee the idividua sequeces. Two simpe exampes wi be ivestigated i Sectio 3, ad Sectio 4 wi focus o the Fiboacci famiy defied i 8. Some geeraizatios wi be discussed at the ed. 2 Product represetatio of certai umber sequeces For ay give set of umbers {x, }, we defie X := x,. 10 We wi first express X i terms of the associated umber sequeces X,m defied i 7. Lemma 2. For ay give set of umbers {x, } with x, C,, N, the sum over x, is give by X = 1 1 X, 1 +1, 11! 2 where X,m,m Z deotes the umber sequeces associated with {x, }. Proof. We first costruct a system of equatios by expicity factorizig 7 for successive m [1,]. To that ed, we defie for p [1,] p+1 X p := 1 =1 p+2 2 =1 x,1 x,2 x,p, where the summatio is subject to the costraits i+1 > i for a i,i [1,p]. Specificay, for p = 1, we have X 1 = x,1 = x,1 +x,2 + +x,, for p = 2 X 2 = 1 =1 1 =1 p=1 x,1x,2 2 =1 2 > 1 = x,1 x,2 +x,1 x,3 + +x, 1 x,, 3

4 ad for p = 3 X 3 = 2 1 =1 2 =1 2 > 1 With this, the product i 7 yieds x,1x,2x,3 3 =1 3 > 2 = x,1 x,2 x,3 +x,1 x,2 x,4 + +x, 2 x, 1 x,. X,m = m+x,1 m+x,2 m+x,3 m+x, = mm+x,2 m+x,3 m+x, + x,1 m+x,2 m+x,3 m+x, = m 2 m+x,3 m+x, + mx,1 +x,2 m+x,3 m+x, +x,1 x,2 m+x,3 m+x, = m +m 1 x,1 +x,2 +x,3 + +x, + +x,1 x,2 x,3 x, = m +m 1 X 1 + +X, fromwhichweobtaiform [1,]thefoowigsystemofiearequatiosiX p,p [1,]: X,1 = X X 2 + +X X,2 = X X 2 + +X X,3 = X X 2 + +X. X, = + 1 X X 2 + +X. This system ca be writte i more compact form as X,i i = j=1 a ij X j, 12 where a ij = i j, i,j [1,]. What remais is to sove 12 for X 1 = X. To that ed, we ote that a i = 1, i [1,], which aows us to costruct a ew system of 1 equatios by subtractig successive equatios i 12. We obtai X,i+1 X,i i+1 i = a 1 ij Xj, 13 j=1 where a 1 ij = a i+1,j a ij = i+1 j i j = i+ j 4

5 for i,j [1, 1]. Agai, a 1 i, 1 = 1, i [1, 1], ad we ca further reduce the system 13 by subtractig successive equatios. After m repetitios, we have m m m m 1 m 1 X,i+ 1 m 1 for i,j [1, m], where a m ij = 1 m m m 1 m i+ = i+ j with a m i, m = m! for a i [1, m]. For m = 1, we fiay obtai 1 X, j=1 a m ij X j = 1!X 1. Chagig the summatio variabe +1 i both sums, ad observig that 1 1 = ad = 1! 2 +1 [10, Equatio 1.14], we fiay arrive at 11. Lemma 2 provides, for ay give 1, a expicit represetatio of the sum over x,, Equatio 11, i terms of a fiite iear combiatio of the umber sequeces X,m. With this, we ca immediatey formuate Lemma 3. For ay give set of umbers {x, } with x, C, N ad associated famiy of umber sequeces {X,m },m Z, the sum over x, obeys for ay give N the idetities X = 1 1 X,+m 1 +1 m 15! 2 ad, for m 0, X = 1!m 1 1 X,m 1 +1m Proof. We first prove 15. Let us defie X,m ad X for the set of umbers m +x, as foows: X m,m := m+m +x, X m := m +x, 5

6 for arbitrary m Z. From the first equatio ad defiitio 7, it foows immediatey that X m,m = X,m+m ad X m = m + X, which together with 11 yied 15. The secod reatio 16 ca be show i a simiar fashio. We defie X,m ad X for the set of umbers x, /m,m Z,m 0 as foows: X m,m := X m := m+ x, m x, m, from which foows that X m,m = 1 m mm +x, = 1 m X,mm 17 ad X m = X /m. Usig agai 11, we obtai 16. Lemma 3 is iterestig i various respects. It ot just geeraizes 11, but aso shows that, for a give N, various combiatios of X,m withi a give famiy of umber sequeces must yied the same resut X. This, i tur, aows us to costruct geera reatios betwee X,m, which wi hod for a famiies of umber sequeces {X,m } represetabe i product form 7, ad costitutes the mai resut of this cotributio. We ca formuate Propositio 4. The members X,m of a give famiy of umber sequeces {X,m },m Z ad N, obey the geera recursive reatio X,m+1 = 1 1 X,+m +! 18 1 ad are subject to the idetity 1 m 1 for m 0. 1 X,m = 1 X, m+1! 19 2 Proof. The proof of 18 utiizes 15 for m m +1 ad m m, yiedig X = 1! = 1! 1 X,+m m X,+m ! X,m m +1 6

7 ad X = 1! = 1! 1 X,+m m m, +1X,+m ! X,m +1 respectivey, where i the ast step 1 was used. Subtractig both idetities ad observig that =, we obtai ! from which ! X,+m ! X 1,m +1 1! X,m+1 + = 0, 1 1 X,+m +1 1! X,m+1 + = 0 foows. After a chage of the summatio variabe +1, the ast reatio yieds 18. I a simiar fashio, 16 yieds together with X, = 1 X! 2!m 1,m m, from which 19 foows. Equatio 18 i Propositio 4 provides geera iear recursios i m Z for X,m. The form of these recursios depeds o, ad cotais a icreasig umber of terms for icreasig. Specificay, for ay give, 18 expresses X,m i terms of X,m with m [m,m 1]. Based o these recursios, usig the geeratig fuctio approach, we ca deduce expicit idetities which express X,m ad X, m for m, i terms of X,m with m [0, 1] ad m [ +1,0], respectivey. Coroary 5. For ay give famiy of umber sequeces {X,m }, the foowig idetities hod + m m! X,m = 1 X, + 20 m m! + m X, m = 1 X, + 1 m! 21 m m! for a N ad m Z with m. 7

8 Proof. We start with 18 for m+1 m, i.e., X,m = 1 + X,+m 1 +!, 1 ad defie the geera geeratig fuctio Az := m 0X,m z m 22 for arbitrary z R, z < 1,z 0. Mutipicatio of X,m with z m ad summatio over m yieds,m z m X m = 1 + X,+m 1 z m +! 1 m m z m. 23 For ay give N, the term o the eft-had side equas X,m z m = X, z +X,+1 z +1 + = Az X, z, m ad the secod term o the right-had side simpifies to! m z m =!z +z +1 + =! z 1 z. To treat the first term o the right-had side, we first exchage the order of both sums, ad observe that 2 X,+m 1 z m = X, 1 z +X, z +1 + = Az X,m z z m +1, m which ca easiy be show by geeraizig the resut for cosecutive 2. For istace, for = 1, we have X,m z m = X,0 z +X,1 z +1 + = Azz, m ad for = 2 X,m +1 z m = X,1 z +X,2 z +1 + = Az X,0 z 1. m With this, 23 takes the form Az X, z = 1 Azz + = Az 8 2 m=0 m=0 X,m z m z +1 +! z 1 z,

9 from which we obtai Az 1 1 z 1 z =2 2 = X, z 1 + X,m z m+ +1 +! 1 =2 m=0 z 1 z. 24 The term o the eft-had side i the ast equatio ca be further simpified by observig that 1 1 z 1 + z +1 1 =2 = z +1 1 = z z z = z z z = 1 1 z +1 1 = 1 z = 1 z, where i the third step we reversed the order of the terms i the sum. Isertig the ast reatio back ito 24 yieds Az = z 2 X, 1 z =2 m=0 1 + z X m+ +1,m 1 1 z +! z 1 z We ow deveop the z-terms i a power series aroud z = 0. Specificay, for the first term 9

10 i 25, we have z 1 z = I a simiar fashio, oe obtais = 1 d 1 1 1! z dz 1 1 z 1 d 1 1! z dz 1 m 0 z m = m! 1 m +1! m 1 = q + 1 z q. 1 q z m z = q m+ +1 1! zm ++1 q m+ 2 1 z q ad z 1 z +1 = q q z q. With this, 25 takes the form Az = q + 1 X, z q 26 1 q 2 =2 m=0 q m+ +1 q m X,m z q +! q q z q, which, we reca, hods for ay N, > 0. What remais is to reorder the terms i the above sums, ad coect a terms of equa power i z. The first term i 26 yieds q + 1 X, z q 1 q q q + 1 = X, z q + 1 q=0 q 1 q + 1 X, z q. 1 Simiary, observig that the secod term i 26 vaishes for = 1 ad yieds a power series 10

11 with miimum degree of 1 for 2, we obtai 2 =2 m=0 q m+ +1 = q m X,m z q 1 1 q +m 1 q +m 2 1 X +m, m 1 1 q=1 m= q+1 + m 2 q +m 2 1 X +m, z q m 1 1 q m=2 The first term i the ast reatio ca further be simpified by chagig the summatio variabe m m +q 1 ad coectig a terms X, for a give : m= q+1 = q +m 1 1 +m m 1 m 1 m q m+ q q 1 X, 1 m q m+ q m= q + 1 X,. 1 q 1 m=0 q 1 = = q 1 Here, we utiized i the ast step the biomia idetity k=0 q +m 2 X, 1 m+ 1 X, 1 m+ 1 1 x k +x 1 1 k = 0, 27 k x 1 which ca easiy be show by iductio i the upper summatio imit ad usig Bri s sum formua [10, Equatio 3.181]. z q 11

12 Now, coectig a terms for q < ad q, we ca the rewrite 26 as Az q q 1 q + 1 q + 1 = X,0 + X,z q + q!z q 1 1 q=1 q + q + 1 m 2 q +m 2 1 +m X,z q 1 m 1 1 q m=2 q = X,q z q +!z q q=0 q + q q +m X, X, 1 +m z q 1 m+1 1 q = q X,q z q +!z q q=0 q q 2 X, q + m= q X, z q, 28 q where i the peutimate step we, agai, coected terms X, for ay give by reorderig the sum, ad i the ast step we utiized the biomia reatio 2 q +m q m = 1 + q. m q m= Notig that q = +a with a N,a 0, this reatio is a direct cosequece of the biomia idetity x k +x 1+a 1 k = a x x x 1+a, 29 k x 1 x+a x 1 k=0 whichitsefisageeraizatioof27adcabeshowbyiductioitheuppersummatio imit usig agai Bri s sum formua [10, Equatio 3.181] ad the biomia idetities = +1 k k k 1 From 28, we fiay obtai Az = k k q=0 X,q z q + q 1 = +1 k q +1 k q. X, + q! z q,

13 which, after compariso with 22, yieds 20. Equatio 21 ca be show i a simiar fashio. Equatio 18 aso aows to deduce a umber of geera idetities the umber sequeces X,m of ay give famiy must obey. Specificay, we have Coroary 6. For,p,q N, p + 1 with p 1, 0 q < p ad m Z, the sequeces X,m of ay give famiy of umber sequeces {X,m } are subject to the foowig idetities: 1 q X p,m + = p X p,m + 1! = Proof. We first show that 30 is vaid for the specia case q = 0 by iductio i p. From 18, after chage of the summatio variabe + 1, we have for 1 1 ad arbitrary m X 1,m ! = 0, ad for m m Subtractig the ast two idetities yieds X 1,m ++1 X 1,m X 1,m + + 1! = 0. 1 X 1,m X 1,m + = 0. Usig =, we obtai X 1,m + = 0, which proves 30 for the specia case q = 0 for p = 1. I a simiar fashio, assumig 30 is true for q = 0 ad a give p 1, we subtract the resutig reatios for 1, m arbitrary, ad m m 1, ad obtai 1 X p+1,m X p+1,m + X p+1,m = 0, 13

14 which yieds X p+1,m + = 0 ad, thus, proves 30 for q = 0 ad p+1. Idetity 30 for 0 < q < p ca be show by iductio i q. Assumig that 30 is true for a give q 0 ad a p q +1, we obtai for 1 ad m m q X p,m + = 0. Usig the biomia idetity 1 = 1 q X p,m + 1 ad further simpified to from which fiay, the ast reatio ca be rewritte as 1 q+1 X p,m + = 0 1 q X p,m 1 1 q+1 X p,m + = 0, 1 q+1 X p,m + = 0 foows, thus provig 30 for q < p. Idetity 31 ca be show i a equivaet fashio through iductio i p, utiizig 18 ad 30. We fiay ote that the recursive reatio 18 ad idetities isted i Coroaries 5 ad 6 are geera ad hod for each famiy of umber sequeces {X,m }, thus suggestig that a famiies costructed from umber sequeces of the form 11 are govered by idetica reatioships betwee their members. I the ext sectio, we wi eaborate o this property, ad briefy cosider two simpe exampes, before iustratig the appicatio to Fiboacci umbers i Sectio 4. 14

15 Tabe 1: The first members of the famiy of power sequeces X,m = m, N ad m Z, Equatio 32 with c = 0. m Two simpe exampes of famiies of umber sequeces 3.1 The famiy of power sequeces Let x, = c = C. I this case, we have X,m = X = m+c = m+c 32 c = c. 33 The first few members of this famiy, for c = 0, are isted i Tabe 1. With the resuts preseted i the previous sectio, we ca immediatey formuate Coroary 7. The famiy of power sequeces X,m = m+c obeys for a c C m+c = m+2c++1! m+c = m 1 2c+m+1! 1 m+c+1 =! m+c m 1 +c = m 1 1 m+1! 15

16 for a N ad m Z, m! 1 m c+ = 1 c+m 1! m! m! 1 m c = 1 c m 1! m! for a,m N with m, ad 1 q m+c p = 0 1 p m+c p =! for a,p N with p+1,p 1 ad m Z. for 0 q < p Proof. A idetities i Coroary 7 are a direct cosequece of Lemmata 2 ad 3, Propositio 4 ad Coroaries 5 ad 6, usig 32 ad 33. We ote that Coroary 7 yieds a umber of iterestig combiatoria, i particuar biomia, idetities ad their geeraizatios. Specificay, for c = 0, Goud 1.13, 1.14 ad 1.47 are recovered [10]. Furthermore, for ay fixed, X,m yieds the sequece of th powers of subsequet itegers m. The third reatio i Coroary 7, for c = 0, provides the the geera form of the th-order iear homogeeous recursios i m with costat coefficiets for such sequeces: m+1 = 1 m +! m Z. For exampe, restrictig to m 0, we obtai for = 2 the sequece of square umbers a m = m 2, obeyig the kow iear recursio a 0 = 0,a 1 = 1,a m+1 = 2a m a m 1 +2,m 1 A000290, M. Kristof, 2005, for = 3 the sequece of cubes a m = m 3, obeyig a 0 = 0,a 1 = 1,a 2 = 2 3,a m+1 = 3a m 3a m 1 +a m 2 +6,m 2 A000578, A. Kig, 2013, ad for = 4 the sequece a m = m 4, subject to the 4th-order iear recursio a 0 = 0,a 1 = 1,a 2 = 2 4,a 3 = 3 4,a m+1 = 4a m 6a m 1 +4a m 2 a m 3 +24,m 3 A000583, A. Kig,

17 3.2 The famiy of Pochhammer umbers Let x, = N. I this case, we have X,m = X = m+ = m+1 35 = 1 +1, 36 2 where a = Γa + /Γa deotes the Pochhammer symbo. The first members of this famiy of sequeces are visuaized i Tabe 2. With the resuts preseted i the ast sectio, we ca immediatey formuate Coroary 8. The famiy of Pochhammer sequeces X,m = m+1 obeys 1 +m = 1 m+! 1 m +1 = m 1+m+1! 1 +m +1 = 1! 1 m +1 m +1 = m 1 m+1! for a N ad m Z, m! 1 m 1+ = 1 1+m 1! m! m! 1 m 1 = 1 1 m 1! m! for a,m N with m, ad 1 q m ++1 p = 0 1 p m ++1 p =! for a,p N with p+1,p 1 ad m Z. 17 for 0 q < p

18 Tabe 2: The first members of the famiy of Pochhammer sequeces X,m = m+1, N,m Z, Equatio 35. m Proof. A idetities i Coroary 8 are a direct cosequece of Lemmata 2 ad 3, Propositio 4 ad Coroaries 5 ad 6, usig 32 ad 33 ad the Pochhammer idetity m+1 = m +1 /m, vaid m Z. As i the case of the famiy of power sequeces, for ay give, the reatios isted i Coroary 8 provide iks betwee Pochhammer umbers m for differet m. Specificay, the third idetity yieds the geera iear recursive rue for sequeces defied by m+1 for ay fixed, amey m+1 = 1 m +!, which is vaid m Z. Restrictig agai to m 0, = 2 yieds the sequece of Obog umbers a m = mm+1 A002378, subject to the recursio a 0 = 0,a 1 = 2,a m+1 = 2a m a m 1 +2,m 1, for = 3 we obtai the sequece a m = mm+1m+2, obeyig a 0 = 0,a 1 = 3!,a 2 = 4!,a m+1 = 3a m 3a m 1 +a m 2 +6,m 2 A007531, Z. Seidov, 2006, ad for = 4 the sequece of products of four cosecutive itegers a m = mm+1m+2m+3 A with a 0 = 0,a 1 = 4!,a 2 = 5!,a 3 = 1 2 6!,a m+1 = 4a m 6a m 1 +4a m 2 a m 3 +24,m 3. We ote that aready for = 2 ad = 4, the recursios obtaied here differ i form from those provided i OEIS [11] for the correspodig sequeces. 18

19 4 The famiy of k-geeraized Fiboacci umbers I the remaider of this cotributio, we wi appy the geera resuts preseted i Sectio 2 to a ess trivia case, amey the geeraized Fiboacci sequeces defied i 9. To that ed, we set π x, = 2icos, from which, usig 8, immediatey foows that X,m = π m 2icos = F m Furthermore, otig that cos π +1, [0,] are the zeros of Chebyshev poyomias of the secod kid 1 U x = 2 x+ +1 x 1 2 x +1 x 2 1 x 2 1 see, e.g., [13, Chapter 22], [14, 8.94], we have X = 2i π cos = 0, where the orthogoaity reatios for Chebyshev poyomias were used. The first members of the famiy of sequeces formed by 39 are visuaized i Tabe 3. Specificay, the secod coum m = 1 cotais the origia Fiboacci sequece F A000045, Equatio 1, ad the third coum m = 2 the sequece of Pe umbers P A000129, Equatio 2, for 1. Whie idividua coums yied subsequet sequeces of geeraized Fiboacci umbers, each row, for fixed, geerates ew iteger sequeces whose eemets are a geeraized Fiboacci umbers. Specificay, for = 2, we obtai, for m 0, the sequece a m = m 2 +1 A002522, for = 3 the sequece a m = m 3 + 2m A ad for = 4 the sequece a m = m 4 +3m 2 +1A Igeera, foraygive N, itegersequecesareobtaied whose expicit form is give i terms of Fiboacci poyomias see, e.g., [6], poyomias of th order i the sequece idex m of the form From Lemmata 2 ad 3, we ca immediatey formuate 2 a m = m 2 = F m

20 Tabe 3: The first members of the famiy of geeraized Fiboacci umbers X,m = F m +1, N,m Z, defied expicity i 39 ad subject to the recursive reatio 9. m Propositio 9. The famiy of geeraized Fiboacci sequeces {F m }, defied i 39, obeys the foowig idetities: 1 1 for a N ad m Z. F +m +1 = m++1! 42 F m +1 = m +1! 43 Proof. Both idetities are a direct cosequece of 15 ad 16, usig 39 ad 40. Equatio 43 ca be used to ik geeraized Fiboacci umbers F m for positive ad egative m. Specificay, subtractig 43 for a give m ad m m, we obtai 1 F m +1 F m +1 = 1 2 m 1 1+1! 44 which, for m = 1, yieds 1 F +1 F +1 = 1 { 0, for eve; ! = +1!, for odd. 45 Furthermore, appicatio of Propositio 4 to the famiy of Fiboacci sequeces eads to 20

21 Propositio 10. The famiy of Fiboacci sequeces F m obeys N the foowig reatios 1 F +m = 1! 46 1 for a m Z. F m +1 m 1 F +1 = m 1 1 m+1! 47 Proof. Both idetities are a cosequece of 18 ad 19, usig 39 ad 40. Reatio 47 ca be directy obtaied aso from 43. We ote that equatio 47 is a specia appicatio of 43, which for m = 1 yieds 1 F F +1 = +1! 48 N, compemetig 45 above. Iterestigy, 46 aows to costruct, for ay give, geera recursive reatios i m for geeraized Fiboacci umbers F m, amey F m+1 = 1 +1 F m +!, 49 which is vaid m Z. Restrictig to m 0, the resutig sequece a m = F m+1 2 = m 2 +1 obtaied for = 2 from 46, obeys a 0 = 1,a 1 = 2,a m+1 = 2a m a m 1 +2,m 1 A002522, E. Werey, Simiary, for = 3, the iteger sequece give by the thirdorder poyomia a m = F m+1 3 = m 3 +2m A054602, obeys the recursive reatio a 0 = 0,a 1 = 3,a 2 = 12,a m+1 = 3a m 3a m 1 +a m 2 +6,m 2, ad for = 4, the sequece a m = F m+1 4 = m 4 + 3m A is subject to the recursio a 0 = 1,a 1 = 5,a 2 = 29,a 2 = 109,a m+1 = 4a m 6a m 1 +4a m 2 a m 3 +24,m 3. Fiay, Coroaries 5 ad 6 provide expicit represetatios of geeraized Fiboacci umbers i terms of other members of the Fiboacci famiy: 21

22 Propositio 11. Geeraized Fiboacci umbers F m obey F m m +1 = 1 1 F m +1 = 1 m 1 1 m F +1 + m! m! m F m! m! 50 51,m N with m, ad F m p+1 = 1 +1 q F m p+1 = 1 +1 p 1 1 for a,p N with p+1,p 1, 0 q < p ad m Z. q F m + p+1 52 p F m + p+1 + p! 53 Proof. The first two idetities are a direct cosequece of 20 ad 20, the ast two ca be show with 30 ad 31, usig 39 ad 40. We ote that Propositios 9 to 11 provide a umber of idetities which iterik the set of geeraized Fiboacci sequeces defied i 39. Specificay, the defiig recursive reatio i, Equatio 9, for geeraized Fiboacci umbers aows to express each F m i terms of F m, < for fixed m, whereas reatios 42 43, ad aow to express each F m i terms of F m, m m for ay give. Combiig both sets of idetities, we arrive at ikig a members of the famiy of geeraized Fiboacci umbers. 5 Cocudig remarks I this cotributio, we ivestigated geera properties of umber sequeces geerated through products of the form 7. We foud severa idetities ad recursive reatios which iterik such sequeces ad suggest their cassificatio i terms of famiies Defiitio 1. Athough the famiies studied here as exampes describe differet iteger sequeces, such as Pochhammer umbers, powers of itegers or geeraized Fiboacci umbers, we fid that each of these famiies is subject to the same set of idetities which, i some cases, geeraize iterestig reatios betwee these kow sequeces. The exampes preseted here costitute but a sma set of potetia appicatios. For istace, q-pochhammer sequeces ad sequeces produced by products of Pochhammer umbers are obtaied for x, = a,a R ad x, = a,a Z, respectivey. The geera reatios isted i Sectio 2 appy i these cases, ad provide a umber of idetities obeyed 22

23 by the correspodig sequeces. By settig x, = 2 qcos π, +1 we obtai with 6 geera Lucas sequeces A108299, i.e., X,m = 1 q m 2 π qcos = L m,q For appropriate m ad q, iterestig idetities, such as gradma s idetity [12], siged bisectios of Fiboacci sequeces ad reatios iterikig powers of Fiboacci umbers, are obtaied. The study of these reatios, their potetia geeraizatio ad appicatio to other famiies of umber sequeces might provide ove ad potetiay usefu isights ito properties shared by quaitativey differet umber sequeces. 6 Ackowedgmet Research supported i part by CNRS. The author wishes to thak L. E. Muer II, J. A. G. Wiow ad S. Hower for vauabe commets. Refereces [1] D. E. Rutherford, Some cotiuat determiats arisig i physics ad chemistry, Proc. Roy. Soc. Ediburgh Sect. A , [2] D. E. Rutherford, Some cotiuat determiats arisig i physics ad chemistry II, Proc. Roy. Soc. Ediburgh Sect. A , [3] D. Lid, Probem H-64, Fiboacci Quart , 116. [4] D. Zeiti, Soutio to probem H-64, Fiboacci Quart , [5] L. Shapiro, Soutio to probem B-742, Fiboacci Quart , [6] R. Adré-Jeai, A ote o a geera cass of poyomias, Fiboacci Quart , [7] R. J. Hede ad C. K. Cook, Recursive properties of trigoometric products, i G. E. Bergum, A. Phiippou ad A. F. Horadam, eds.,appicatios of Fiboacci Numbers, Vo. 6, Spriger, 1996, pp [8] J. Ciger, q-fiboacci poyomias, Fiboacci Quart ,

24 [9] Z. H. Su, Expasio ad idetities cocerig Lucas sequeces, Fiboacci Quart , [10] H. W. Goud, Combiatoria Idetities, Morgatow, [11] N. J. A. Soae, The o-ie ecycopedia of iteger sequeces, [12] S. Humbe, Gradma s idetity, Math. Gaz , [13] M. Abramowitz ad I. A. Stegu, Hadbook of Mathematica Fuctios with Formuas, Graphs, ad Mathematica Tabes, Dover, [14] I. S. Gradshtey ad I. M. Ryzhik, Tabe of Itegras, Series, ad Products, Esevier, Mathematics Subject Cassificatio: Primary 11B39; Secodary 32A05, 40B05. Keywords: product represetatio, Fiboacci umber, Pe umber, Pochhammer umber, iteger power, recursive idetity. Cocered with sequeces A000045, A000129, A000290, A000578, A000583, A002378, A002522, A007531, A052762, A054602, A057721, ad A Received September ; revised versio received March Pubished i Joura of Iteger Sequeces, Apri Retur to Joura of Iteger Sequeces home page. 24