h-analogue of Fibonacci Numbers

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1 h-aalogue of Fboacc Numbers arxv: v [math-ph 30 Sep 009 H.B. Beaoum Prce Mohammad Uversty, Al-Khobar 395, Saud Araba Abstract I ths paper, we troduce the h-aalogue of Fboacc umbers for o-commutatve h-plae. For hh ad h 0, these are just the usual Fboacc umbers as t should be. We also derve a collecto of dettes for these umbers. Furthermore, h-bet s formula for the h-fboacc umbers s foud ad the geeratg fucto that geerates these umbers s obtaed. 000 Mathematcal Subject Classfcato : B39, B65, B83, 05A30, 05A0 Keyword : Mathematcal physcs, No-Commutatve Geometry, Geeralzed Fboacc umbers ad Polyomals, Bomal Coeffcets To my wfe Nawal Emal : hbeaoum@pmu.edu.sa, hbeaoum@physcs.syr.edu

2 Itroducto Fboacc recursve sequece has fascated scholars ad amateurs for cetures. Sce ther appearace the boo Lber Abac publshed 0 by the Itala medeval mathematca Leoardo Fboacc, they have bee ecoutered may dstcts cotexts, ragg from the arts, pure mathematcs ad physcal sceces to electrcal egeerg. These umbers are ot radom umbers but each umber s made by addg the prevous umber to the preset oe. The rato of successve pars teds to the so-called golde rato ϕ ad whose recprocal s , so that we have ϕ +ϕ. Ths golde rato s a rratoal umber wth several curous propertes ad oe ca come across ths rato may areas of arts ad sceces. I fact, the acet Grees foud t a very terestg ad dve umber. Bet s formula for Fboacc umbers s exceptoal because t s expressed terms of rratoal umber, eve though all Fboacc umbers are tegers. Fboacc umbers have bee studed both for ther applcatos ad the mathematcal beauty of the rch ad vared dettes that they satsfy. The referece [ cotas may results o Fboacc umbers wth detaled proofs. Iterestgly eough, the amazg Fboacc umbers seem to be trsc ature sce they have bee detfed leaves, petal arragemets, pecoes, peapples, seeds ad shells. The Fboacc sequece s defed by the recurrece : wth the tal codtos f 0 0 ad f. f + f + f. It s easy to deduce the followg detty that coects Fboacc umbers f ad the bomal coeffcets : f [ 0 Smlarly, the Fboacc polyomals are defed as : f u,v [ 0, > 0. u v.3 The ma objectve of ths paper s to geeralze the results o classcal Fboacc umbers to o-commutatve h-plae. That s to troduce the h-aalogue of Fboacc umbers o the o-commutatve h-plae usg the h-bomal coeffcets [. I secto, we defe the h-pascal tragle whch arses aturally from

3 the h-bomal coeffcets ad prove some elegat dettes whch we relate to the Charler polyomals. Secto 3 focuses o the h-fboacc umbers by provdg the ecessary defto ad shows that the ow Fboacc dettes ad ther bjectve proofs ca be easly exteded to bjectve proofs of h-aalogues of these dettes. Secto 4 s devoted to reformulate the h-fboacc sequece terms of a matrx represetato. h-bet s formula for the h-fboacc operators s troduced secto 5 ad a umber of dettes are derved. Fally, the geeratg fucto of the h-fboacc sequece s obtaed the last secto. Ths secto also provdes us wth lsts the geeratg fuctos for the varous powers ad products of h-fboacc sequece. I [, we troduced the h-aalogue of Newto s bomal formula : [ x + y y x.4 h,h 0 where x ad y are o-commutg varables satsfyg : Here [ wth hh. xy yx + hy.5 s the h-bomal coeffcets gve as follows : h,h [ h h ; h,h hh.6 h; a s the shfted factoral defed as : a s; { 0 aa + s... a + s,,....7 I what follows, we cosder the h-bomal coeffcets wth two parameters h ad h such that hh s ot ecessarly equal to. These coeffcets obey to the followg propertes : ad [ + [ + + h,h [ h,h + hh h,h hh + + [ [ h,h + h,h

4 h-pascal tragle The h-pascal tragle s costructed by cosderg the h-bomal coeffcet of the th colum 0,,,3, ad the th row 0,,,3,. \ hh h; hh h; hh h; hh h; 4 hh h; hh h; hh h; 0 hh h; 5 hh h;3 hh h; hh h; 0 hh h; 5 hh h;3 6 hh h;4 hh h;5 7 7 hh h; 35 hh h; 35 hh h;3 hh h;4 7 hh h;5 hh h;6 Table : h-pascal tragle If we sum the h-bomal coeffcets of the h-pascal tragle, the followg detty that coects the h-bomal coeffcets to Charler polyomals s obtaed : [ c h,h. 0 h where c z,a are the Charler polyomals defed by the followg formula [3 : c z,a 0 a z ;. Aother detty satsfed by the h-pascal Tragle s the sum of all elemets a colum s gve by : hh + j [ j 3 h-fboacc umbers h,h [ + j + h,h.3 I ths secto, the h-fboacc umbers are troduced. It should be oted that the recurrece formula of these umbers depeds o the parameters h ad h. They reduced to the usual Fboacc umbers whe hh ad h goes to zero. 4

5 The h-fboacc umbers whch are obtaed by addg dagoal umbers of the h-pascal tragle, are gve by : F h,h [ 0 [ h,h, > 0 3. Sce hypergeometrc fuctos are mportat tool may braches of pure ad appled mathematcs, a drect coecto betwee h-fboacc umbers ad hypergeometrc fuctos ca be establshed. Ideed we have : F h,h 3 F / + /, / +,h ; + ; 4h 3. The lst the frst 0 h-fboacc, whe expaded powers seres of h ad h, are show the table below : F h,h f hh 4 + hh hh + h h h hh + 3h h h hh + 6h h h + + h 3 h h + h hh + 0h h h + + 4h 3 h h + h hh + 5h h h + + 0h 3 h h + h + + h 3 h h + h + h hh + h h h + + 0h 3 h h + h + + 5h 4 h h + h + h Table : Frst 0 umbers of h-fboacc ad Fboacc umbers. We also provded the table a lst of the classcal Fboacc umbers just to compare both of them. As t was atcpated, the F h,h sequece reduces to the usual f whe hh ad h goes to zero. Theorem. h-fboacc recurrece The h-fboacc umbers obey the followg recurrece formula F h,h + F h,h + hh F h,h

6 Usg equato 7., we have : F h,h + [ 0 [ F h,h + hh [ h,h 0 [ 0 [ [ h,h + h,h + hh [ [ F h,h + hh F h,h + h,h + As the usual Fboacc umbers, the h-fboacc umbers satsfy umerous dettes. We express some of them below : Theorem. The h-fboacc umbers have the property : hh F h,h + F h,h Usg the h-fboacc recurrece relato, we have : hh F h,h + F h,h + F h,h + hh F h,h + F h,h+ + F h,h hh F h,h + F h,h F h,h. hh F h,h + 3 F h,h 5 F h,h 4 hh F h,h + F h,h + 4 F h,h + 3 hh F h,h + F h,h 3 F h,h Addg all these equatos, we get : hh F h,h + F h,h + + F h,h + F h,h + Theorem 3. The followg detty holds h h ; F h,h + F h,h 3.5 6

7 Usg the h-fboacc recurrece relatos, we have : F h,h F h,h hh F h,h + F h,h + 3 F h,h + hh + F h,h + 4 F h,h + 5 F h,h + 4 hh + F h,h +3 6 F h,h +. F h,h +. hh + F h,h + + F h,h F h,h hh + 3 F h,h + 4 F h,h + 3 F h,h + 4 hh + F h,h + F h,h + F h,h + hh + F h,h + 0 Multplyg F h,h + by h h ; ad addg these equatos, we get : h h ; F h,h + F h,h h h ; F h,h ++ 0 F h,h Theorem 4. For > 0, we have the followg property of the h-fboacc umbers, h h ; F h,h + F h,h + h h ; 3.6 Usg the h-fboacc recurrece relatos, we have : F h,h F h,h + hh F h,h + F h,h + F h,h + hh + F h,h + 3 F h,h + 4 F h,h + 3 hh + F h,h +3 5 F h,h +. F h,h + + hh + F h,h + +. F h,h + 4 F h,h + 5 hh + F h,h + 3 F h,h + F h,h + 3 hh + F h,h + 7

8 Multplyg F h,h + by h h ; ad addg these equatos, we get : h h ; F h,h + F h,h + h h ; F h,h + F h,h + h h ; Aother way of troducg the Fboacc umbers s to use the Q-matrx formulato where Q s gve by : Q 0 Now by rasg Q to the th power, t ca be show that : Q f+ f where ±, ±, ±3,. f f I the ext secto, we wll troduce the h-fboacc matrces based o h- Fboacc operators. Here the Q h -matrx operators are utlzed whch are a geeralzato of the Q-matrx that depeds o the parameter h. 4 Matrx Represetato of h-fboacc Numbers Sce the h-fboacc umbers volve the shfted factoral, t s coveet for us to use repeated dervatos to hadle t : h d dt t h t h h ; 4. The latter equato permts us to defe what we call here the h-fboacc operators as follows : where F 0 0,F. F [ 0 h d dt 4. Wth these operators, the h-fboacc umbers ca be expressed as : F h,h F t h t 4.3 8

9 Moreover, t s easy to see that the Fboacc operators obey the followg recurrece formula : F + F h d dt F 4.4 Ths sequece of operators ca be exteded to egatve subscrpts by defg them as : h d dt F F + F 4.5 To reformulate the h-fboacc umbers a matrx represetato, let use frst cosder the matrx operators Q h, Q h h d 4.6 dt 0 whch ca be represeted terms of the h-fboacc operators as follows, F Q h F h d dt F h d dt F 0 The geeral, for the th power of the Q h -matrx, we wll get : F+ F Q h h d dt F h d dt F Theorem 5. For ay gve > 0, the followg property holds for the th power of the Q h,h - matrx : Q h,h where Q h,h Q h t h t F h,h + F h,h hh F h,h + hh F h,h + The proof of ths theorem s straghtforward usg deftos. By tag the verse of the Q h -matrx, t s easy to fd that : h d h d dt Q h dt F 0 F h d dt F F ad geeral, the verse of the Q h -matrx to the th power ca be wrtte as : h d dt Q h h d dt F F h d dt F 4. F + whch meas that : h d dt F F 4. Ths result mples that the h-fboacc umbers wth egatve dces ca be expressed terms of the postve dces. 9

10 Theorem 6. For 0, we have the followg property that relates the h-fboacc umber wth egatve dex to the oe wth a postve dex, hh F h,h + F h,h 4.3 By defto, we have h d F t h t hh h,h F + dt Now usg equato 4.3 we get, hh F h,h + F t h t F h,h Next we derve some ce dettes betwee h-fboacc operators. Theorem 7. Let be a postve teger. The F + F F h d dt 4.4 Ths theorem s easly prove by tag the determat equato 4.9 ad usg the fact that detq h detq h. Theorem 8. Let ad m be postve tegers. The we have, F m++ F m+ F + h d dt F mf F m+ F m+ F h d dt F mf F m+ F m F + h d dt F m F F m+ F m F h d dt F m F 4.5 The proof of ths theorem s straghtforward by usg that Q h m+ Q h m Q h ad equatg the correspodg matrx etres. 0

11 Sce Q h -matrx s a matrx, the matrx powers of ths matrx are ot depedet. Ideed the Cayley-Hamlto theorem mples that : Hece : Q h Q h h d dt I 4.6 Q h F Q h h d dt F I 4.7 Theorem 9. We have 0 h d F dt F F F 4.8 By tag the th power of equato 4.7, we get Q h F Q h h d dt F I 0 0 F h d F dt Q h F h d F dt F Q h h ddt I By equatg the matrx elemet above, we get : F 0 h d F dt F F Smlarly, usg the Cayley-Hamlto theorem we ca re-express equato 4. as : h d Q dt h + F Q h F + I 4.9 Theorem 0. We have 0 + F F + F F 4.0

12 By tag the th power of equato 4.9, we get h d Q dt h + F Q h F + I + 0 F + 0 The matrx elemet of the above matrx gves : + F 0 F F + Q h F Q h h d dt F I F F + F whch meas F 0 + F F + F 5 h-bet Formula Bet s formula s well ow the Fboacc umbers theory. I ths secto, we derve the h-bet s formula for the h-fboacc umbers usg the Q h -matrx formulato. Theorem. for all 0, we have for the h-fboacc operators F λ + λ where λ ± are the egevalues of the matrx Q h, h d dt gve by : λ ± ± + 4 h d dt The egevalues of the matrx Q h are obtaed from : λ λ 0 h d dt 5. 5.

13 By solvg the determat for λ, we get the two real solutos λ ± equato 5.. Now the matrx Q h ca be wrtte terms of the egevalues λ ± ad egevectors as : Q h λ λ + λ+ 0 0 λ λ λ + From ths, we obta the Q h matrx to the th power, λ Q h + 0 λ λ + λ λ + 0 λ λ + + λ+ λ+ λ λ + λ λ + λ λ + λ λ + λ λ+ λ We fally get : F q + λ + λ q +4 h d dt + 4 h d dt +4 h d dt We use the Bet s formula to derve some dettes betwee h-fboacc operators ad umbers. Theorem. h-catala s detty The followg property holds for h-fboacc operators : F m F +m F + m h d m F m dt 5.3 3

14 Usg h-bet s formula, we have : F m F +m F λ m + λ m. λ+m + λ +m λ + λ λ m + λ+m λ+ +m λ m + λ + λ Now sce λ + λ h d dt, we get λ +λ λ m + λ + m λ + λ λ +λ m λ m + + λ m λ m +λ m λ + λ m λ m + λ m λ + λ m F m F m F +m F + m h d dt m F m Theorem 3. h- d Ocage detty If > m, the F m F + F m+ F h d dt F m Usg aga the h-bet s formula, we have : + λ+ F m F + F m+ F λm + λm. λ+ + λ m+. λ + λ λm+ λ+ + λm λm + λ+ + λ + λm+ + λ m+ + λ λ +λ m + λ m +λ λ + + λ λ + λm + λm + λ λ + λ λm + λ m h dt d F m 5.4 4

15 Theorem 4. h d dt F F 5.5 By usg the h-bet s formula, we have : h d dt F λ + λ λ + λ λ + λ + λ + λ + λ + λ + λ λ λ + λ + λ + λ λ + + λ + λ F λ + λ λ where we have used λ ± λ. Theorem 5. F h,h + F h,h 5.6 Ths theorem s easly prove usg the latter theorem. Theorem 6. F F 5.7 5

16 F λ + λ + λ + λ λ λ + λ + λ F λ + λ Theorem 7. F h,h F h,h 5.8 The proof follows from the latter theorem. 6 Geeratg Fucto for h-fboacc Operators I ths secto, the geeratg fuctos for the h-fboacc operators are gve. As a result, the h-fboacc operator sequeces are see as the coeffcets of the power seres of the correspodg geeratg fucto. To derve a geeratg fucto for h-fboacc operators, cosder the fucto gx gve by : gx F x 6. 0 It follows that gx F 0 x 0 F x gx x 6

17 Hece gx x From whch we get : Thus x gx F x F x + h ddt F x F x λ + λ x F x where we have used λ + + λ. x gx λ + λ x gx 0 x + λ+ λ x gx x x x + λ + λ x x λ + x λ x So the geeratg fucto for the h-fboacc operators s : gx x λ + x λ x F x 6. Next we lst the geeratg fuctos that geerate the varous powers ad products of the h-fboacc sequeces. Theorem 8. We have 0 x λ + x λ x F x 6.3 Usg the h-bet formula, we have : F x λ + λ x λ + x 0 λ x λ + x λ x λ + λ x λ + x λ x x λ + x λ x 7

18 Theorem 9. We have Usg the h-bet formula, we have : F + x 0 + λ + λ x λ + x λ x F + x 6.4 λ+ + λ x λ + + x 0 λ+ λ + x λ λ x + λ + λ x λ + x λ x + λ + λ x λ + x λ x λ + x Theorem 0. We have F m + λ + λ F m x λ + x λ x Usg the h-bet formula, we have : F m+ x 0 λ m+ + λ m+ λ 0 + λ 0 F m+ x x λ m+ + x 0 λ m + λ + x λm λ x λm + λm + λ +λ λ m + λ m x λ + x λ x F m λ + λ F m x λ + x λ x λ m+ x 8

19 Theorem. We have x + λ + λ x λ + x λ x λ + λ x x λ + x λ x λ + λ x λ + x λ x λ + λ x F x F F + x F + F + x 6.8 x + λ + λ x + λ 3 + λ3 x3 λ 3 + x λ 3 x λ + λ x λ + λ x F 3 x 6.9 Easy to prove usg h-bet s formula. The followg proposto gves us the value for the h-fboacc sequece seres wth weghts p +. Theorem. For each o-vashg teger umber p : 0 0 F p + Usg the h-bet s formula, we get : F + p + λ p +.λ 0 p 0 0 p p + λ + λ p λ + p λ λ+ p 0 λ p Equato 6.0 yelds the followg results for partcular values of p : Whe p Whe p 3 Whe p 8 Whe p 0 0 F 0 F 0 F λ + λ f ++λ + λ λ + λ f 5 ++λ + λ λ + λ f 0 ++λ + λ F λ + λ f ++λ + λ where f appearg the deomator s the usual Fboacc umber. 9

20 7 Coclusos h-aalogue of Fboacc umbers have bee troduced ad studed. Several propertes of these umbers are derved. I addto, the h-bet s formula for these umbers s foud ad the geeratg fucto of these h-fboacc sequeces ad ther varous powers have bee deduced. It s straghtforward to troduce the h-lucas umbers. Ths wor s progress [4. It s possble to troduce the q h-aalogue of Foacc umbers by usg the q h-aalogue of bomal coeffcets whch was troduced [5. Ideed the q h-aalogue of bomal coeffcets was foud to be : [ [ h [ h ;[ hh 7. h;[ q,h,h These coeffcets obey to the followg propertes : q q ad [ + q,h,h [ + + q [ q,h,h q,h,h hh [ + q [ + q + hh [ [ q,h,h + q,h,h + The q h-aalogue of Fboacc umbers wll be defed as follows : q F h,h + [ 0 q [ ad they obey the followg recurrece formula : q,h,h q F h,h + q F h,h + q q F h,h For hh ad h 0, the q h-aalogue of Fboacc umbers are just the q-fboacc umbers see [6. Smlarly, several propertes of the q h-aalogue of Fboacc umbers ca be derved. We lst below some of them whch are easy to prove from the recurrece formula. 0 0 hh 0 q q F h,h + q F h,h + q h h q ; F h,h + q F h,h h h ; q F h,h + /q q F h,h + h h ; 7.6 0

21 Acowledgmets I would le to tha Tom Koorwder for hs helpful commets ad suggestos. Refereces [ T. Koshy, Fboacc ad Lucas Numbers wth Applcatos, Wley, New Yor, 00. [ H. Beaoum, h aalog of Newto s bomal formula, J.Phys.A:Math. Ge. 3 L75,998. e-prt: math-ph/980. [3 T.S. Chhara, A Itroducto to Orthogoal Polyomals, Gordo ad Breach, New Yor, 978. [4 H. Beaoum, h-aalogue of Lucas umbers, wor progress. [5 H. Beaoum, q,h - aalog of Newto s bomal formula, J.Phys.A:Math. Ge ,999. e-prt: math-ph/9808. [6 L. Carltz, Fboacc otes. 3 : q-fboacc umbers, Fboacc Quarterly ; Fboacc otes 4 : q-fboacc polyomals, Fboacc Quarterly 3 97, 975.