h-analogue of Fibonacci Numbers

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1 h-analogue of Fbonacc Numbers arxv: v [math-ph 30 Sep 009 H.B. Benaoum Prnce Mohammad Unversty, Al-Khobar 395, Saud Araba Abstract In ths paper, we ntroduce the h-analogue of Fbonacc numbers for non-commutatve h-plane. For hh and h 0, these are just the usual Fbonacc numbers as t should be. We also derve a collecton of denttes for these numbers. Furthermore, h-bnet s formula for the h-fbonacc numbers s found and the generatng functon that generates these numbers s obtaned. 000 Mathematcal Subject Classfcaton : B39, B65, B83, 05A30, 05A0 Keyword : Mathematcal physcs, Non-Commutatve Geometry, Generalzed Fbonacc numbers and Polynomals, Bnomal Coeffcents To my wfe Nawal Emal : hbenaoum@pmu.edu.sa, hbenaoum@physcs.syr.edu

2 Introducton Fbonacc recursve sequence has fascnated scholars and amateurs for centures. Snce ther appearance n the boo Lber Abac publshed n 0 by the Italan medeval mathematcan Leonardo Fbonacc, they have been encountered n many dstncts contexts, rangng from the arts, pure mathematcs and physcal scences to electrcal engneerng. These numbers are not random numbers but each number s made by addng the prevous number to the present one. The rato of successve pars tends to the so-called golden rato ϕ and whose recprocal s , so that we have ϕ +ϕ. Ths golden rato s an rratonal number wth several curous propertes and one can come across ths rato n many areas of arts and scences. In fact, the ancent Grees found t a very nterestng and dvne number. Bnet s formula for Fbonacc numbers s exceptonal because t s expressed n terms of rratonal number, even though all Fbonacc numbers are ntegers. Fbonacc numbers have been studed both for ther applcatons and the mathematcal beauty of the rch and vared denttes that they satsfy. The reference [ contans many results on Fbonacc numbers wth detaled proofs. Interestngly enough, the amazng Fbonacc numbers seem to be ntrnsc n nature snce they have been dentfed n leaves, petal arrangements, pnecones, pneapples, seeds and shells. The Fbonacc sequence s defned by the recurrence : wth the ntal condtons f 0 0 and f. f n+ f n +f n. It s easy to deduce the followng dentty that connects Fbonacc numbers f n and the bnomal coeffcents : f n [ n n Smlarly, the Fbonacc polynomals are defned as : f n u,v [ n n, n > 0. u n v.3 The man objectve of ths paper s to generalze the results on classcal Fbonacc numbers to non-commutatve h-plane. That s to ntroduce the h-analogue of Fbonacc numbers on the non-commutatve h-plane usng the h-bnomal coeffcents [. In secton, we defne the h-pascal trangle whch arses naturally from

3 the h-bnomal coeffcents and prove some elegant denttes whch we relate to the Charler polynomals. Secton 3 focuses on the h-fbonacc numbers by provdng the necessary defnton and shows that the nown Fbonacc denttes and ther bjectve proofs can be easly extended to bjectve proofs of h-analogues of these denttes. Secton 4 s devoted to reformulate the h-fbonacc sequence n terms of a matrx representaton. h-bnet s formula for the h-fbonacc operators s ntroduced n secton 5 and a number of denttes are derved. Fnally, the generatng functon of the h-fbonacc sequence s obtaned n the last secton. Ths secton also provdes us wth lsts the generatng functons for the varous powers and products of h-fbonacc sequence. In [, we ntroduced the h-analogue of Newton s bnomal formula : [ x+y n n y x n.4 h,h where x and y are non-commutng varables satsfyng : Here [ n wth hh. xy yx+hy.5 s the h-bnomal coeffcents gven as follows : h,h [ n n h h ; h,h n hh.6 h; a s the shfted factoral defned as : a s;n { n 0 aa+s...a+ns n,,....7 In what follows, we consder the h-bnomal coeffcents wth two parameters h and h such that hh s not necessarly equal to. These coeffcents obey to the followng propertes : and [ n+ [ n+ + h,h [ n h,h +hh h,h hh n+ + [ n [ n h,h + h,h

4 h-pascal trangle The h-pascal trangle s constructed by consderng the h-bnomal coeffcent of the nth column n 0,,,3, and the th row 0,,,3,. n\ 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 trangle If we sum the h-bnomal coeffcents of the h-pascal trangle, the followng dentty that connects the h-bnomal coeffcents to Charler polynomals s obtaned : [ n c n h,h. h where c n z,a are the Charler polynomals defned by the followng formula [3 : c n z,a n a z ;. Another dentty satsfed by the h-pascal Trangle s the sum of all elements n a column s gven by : hh +j [ j 3 h-fbonacc numbers h,h [ n+ j + h,h.3 In ths secton, the h-fbonacc numbers are ntroduced. It should be noted that the recurrence formula of these numbers depends on the parameters h and h. They reduced to the usual Fbonacc numbers when hh and h goes to zero. 4

5 The h-fbonacc numbers whch are obtaned by addng dagonal numbers of the h-pascal trangle, are gven by : F h,h n [ n [ n h,h, n > 0 3. Snce hypergeometrc functons are mportant tool n many branches of pure and appled mathematcs, a drect connecton between h-fbonacc numbers and hypergeometrc functons can be establshed. Indeed we have : F h,h n 3 F n/+/,n/+,h ;n+;4h 3. The lst the frst 0 h-fbonacc, when expanded n powers seres of h and h, are shown n the table below : n F h,h n f n 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 numbers of h-fbonacc and Fbonacc numbers. We also provded n the table a lst of the classcal Fbonacc numbers just to compare both of them. As t was antcpated, the F h,h n sequence reduces to the usual f n when hh and h goes to zero. Theorem. h-fbonacc recurrence The h-fbonacc numbers obey the followng recurrence formula F h,h n+ F h,h n +hh F h,h + n 3.3 5

6 Usng equaton 7., we have : F h,h n+ [ n [ n F h,h n +hh [ n h,h [ n [ n [ n h,h + h,h +hh [ n [ n F h,h n +hh F h,h + n h,h + As the usual Fbonacc numbers, the h-fbonacc numbers satsfy numerous denttes. We express some of them below : Theorem. The h-fbonacc numbers have the property : hh Usng the h-fbonacc recurrence relaton, we have : F h,h + F h,h + n+ 3.4 hh F h,h + n F h,h n+ F h,h n+ hh F h,h + n Addng all these equatons, we get : F h,h+ n+ F h,h n hh F h,h + n F h,h n F h,h n. 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 hh F h,h + F h,h + n+ F h,h + F h,h n+ Theorem 3. The followng dentty holds h n h ;n F h,h +n F h,h n 3.5 6

7 Usng the h-fbonacc recurrence relatons, we have : F h,h n F h,h n hh F h,h + n F h,h + n3 F h,h + n hh + F h,h + n4 F h,h + n5 F h,h + n4 hh + F h,h +3 n6 F h,h +n. F h,h +n. hh +n F h,h +n+ F h,h +n3 5 F h,h +n3 6 hh +n3 F h,h +n 4 F h,h +n 3 F h,h +n 4 hh +n F h,h +n F h,h +n F h,h +n hh +n F h,h +n 0 Multplyng F h,h +n by h n h ;n and addng these equatons, we get : h n h ;n F h,h +n F h,h n h n h ;n F h,h +n+ 0 F h,h n Theorem 4. For n > 0, we have the followng property of the h-fbonacc numbers, h n h ;n F h,h +n F h,h n+ hn h ;n 3.6 Usng the h-fbonacc recurrence relatons, we have : F h,h n F h,h n+ hh F h,h + n F h,h + n F h,h + n hh + F h,h + n3 F h,h + n4 F h,h + n3 hh + F h,h +3 n5 F h,h +n. F h,h +n + hh +n F h,h +n+. F h,h +n 4 F h,h +n 5 hh +n F h,h +n 3 F h,h +n F h,h +n 3 hh +n F h,h +n 7

8 Multplyng F h,h +n by h n h ;n and addng these equatons, we get : h n h ;n F h,h +n F h,h n+ hn h ;n F h,h +n F h,h n+ hn h ;n Another way of ntroducng the Fbonacc numbers s to use the Q-matrx formulaton where Q s gven by : Q 0 Now by rasng Q to the nth power, t can be shown that : Q n fn+ f n where n ±,±,±3,. f n f n In the next secton, we wll ntroduce the h-fbonacc matrces based on h- Fbonacc operators. Here the Q h -matrx operators are utlzed whch are a generalzaton of the Q-matrx that depends on the parameter h. 4 Matrx Representaton of h-fbonacc Numbers Snce the h-fbonacc numbers nvolve the shfted factoral, t s convenent for us to use repeated dervatons to handle t : h d dt t h t h h ; 4. The latter equaton permts us to defne what we call here the h-fbonacc operators as follows : where F 0 0,F. F n [ n n h d dt 4. Wth these operators, the h-fbonacc numbers can be expressed as : F h,h n F n t h t 4.3 8

9 Moreover, t s easy to see that the Fbonacc operators obey the followng recurrence formula : F n+ F n h d dt F n 4.4 Ths sequence of operators can be extended to negatve subscrpts by defnng them as : h d dt F n F n +F n 4.5 To reformulate the h-fbonacc numbers n a matrx representaton, let use frst consder the matrx operators Q h, Q h h d 4.6 dt 0 whch can be represented n terms of the h-fbonacc operators as follows, F Q h F h d dt F h d dt F 0 Then n general, for the nth power of the Q h -matrx, we wll get : n Fn+ F Q h n h d dt F n h d dt F n Theorem 5. For any gven n > 0, the followng property holds for the nth power of the Q h,h - matrx : Q n h,h where Q n h,h Q h n t h t F h,h n+ F h,h n hh F h,h + n hh F h,h + n The proof of ths theorem s straghtforward usng defntons. By tang the nverse of the Q h -matrx, t s easy to fnd that : h d h d dt Q h dt F 0 F h d dt F F and n general, the nverse of the Q h -matrx to the nth power can be wrtten as : h d dt n Q n h n h d dt F n F n h d dt F 4. n F n+ whch means that : h d dt n F n n F n 4. Ths result mples that the h-fbonacc numbers wth negatve ndces can be expressed n terms of the postve ndces. 9

10 Theorem 6. For n 0, we have the followng property that relates the h-fbonacc number wth negatve ndex to the one wth a postve ndex, hh n F h,h +n n n F h,h n 4.3 By defnton, we have h d n F nt h t hh n h,h F +n n dt Now usng equaton 4.3 we get, hh n F h,h +n n n F n t h t n F h,h n Next we derve some nce denttes between h-fbonacc operators. Theorem 7. Let n be a postve nteger. Then F n+ F n F n n h d dt n 4.4 Ths theorem s easly proven by tang the determnant n equaton 4.9 and usng the fact that detq h n detq h n. Theorem 8. Let n and m be postve ntegers. Then we have, F m+n+ F m+ F n+ h d dt F mf n F m+n F m+ F n h d dt F mf n F m+n F m F n+ h d dt F mf n F m+n F m F n h d dt F mf n 4.5 The proof of ths theorem s straghtforward by usng that Q h m+n Q h m Q h n and equatng the correspondng matrx entres. 0

11 Snce Q h -matrx s a matrx, the matrx powers of ths matrx are not ndependent. Indeed the Cayley-Hamlton theorem mples that : Hence : 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 n h d n F dt Fn F F n 4.8 By tang the nth power of equaton 4.7, we get Q n h F Q h h d n dt F I n n 0 0 F h d n F n dt Q h F h d n F n dt F Q h h ddt I By equatng the matrx element above, we get : F n 0 n h d n F dt Fn F Smlarly, usng the Cayley-Hamlton theorem we can re-express equaton 4. as : h d Q dt h + F Q h F + I 4.9 Theorem 0. We have 0 n + F Fn + F F n 4.0

12 By tang the nth power of equaton 4.9, we get h d Q dt h n+n F Q h F + I n+n n 0 n n F n + 0 The matrx element of the above matrx gves : n+ F n n 0 F n F n + Q h n F Q h h d dt F I F Fn + F whch means F n 0 n + F Fn + F 5 h-bnet Formula Bnet s formula s well nown n the Fbonacc numbers theory. In ths secton, we derve the h-bnet s formula for the h-fbonacc numbers usng the Q h -matrx formulaton. Theorem. for all n 0, we have for the h-fbonacc operators F n λ n λ + λ + λ n where λ ± are the egenvalues of the matrx Q h,h d dt gven by : λ ± ± +4h d dt The egenvalues of the matrx Q h are obtaned from : λ h d dt λ

13 By solvng the determnant for λ, we get the two real solutons λ ± n equaton 5.. Now the matrx Q h can be wrtten n terms of the egenvalues λ ± and egenvectors as : Q h λ λ + λ+ 0 0 λ λ λ + From ths, we obtan the Q h matrx to the nth power, n λ n Q h + 0 λ λ + 0 λ n λ n+ + λn+ λ+ λ λ n + λ n λ + λ λ n + λ n λ + λ λ+ n λn We fnally get : F n λ n λ + λ + λ n + q +4h d dt n +4h d dt q n +4h d dt We use the Bnet s formula to derve some denttes between h-fbonacc operators and numbers. Theorem. h-catalan s dentty The followng property holds for h-fbonacc operators : F nm F n+m F n n+m h d nm F m dt 5.3 3

14 Usng h-bnet s formula, we have : F nm F n+m F n λnm Now snce λ + λ h d dt, we get + λ nm. λn+m + λ n+m λ n + λ n λnm + λn+m λ+ n+m λ nm+λ n + λn λ +λ n λ λ + m + λ + λ m λ +λ nm λ m + +λ m λ m +λ m λ + λ nm λ m + λ m λ + λ nm F m F nm F n+m F n n+m h d dt nm F m Theorem 3. h- d Ocagne dentty If n > m, then F m F n+ F m+ F n n h d dt n F mn Usng agan the h-bnet s formula, we have : + λn+ F m F n+ F m+ F n λm + λm. λn+ λm+ + λ m+ λn+ + λm λm + λn+ +λn + λm+ +λ m+ + λn λn +λ m +λ m +λ n λ + +λ λn + λm +λm + λn λ + λ n λmn + λ mn n h dt d n F mn. λn + λn 5.4 4

15 Theorem 4. n h d dt n F F n 5.5 By usng the h-bnet s formula, we have : n h d dt n F n λ + λ n λ + λ n λ + λ n λ + λ + λ +λ + n λ + λ +λ n λ n λ + + n λ n λ + + n λn + λn F n n λ + λ n λ where we have used λ ± λ. Theorem 5. n F h,h +n F h,h n 5.6 Ths theorem s easly proven usng the latter theorem. Theorem 6. n n F n F n 5.7 5

16 n n F n n λ + λ n +λ + n λ n λ n λ + n n λn + λn n F n n λ + n n λ Theorem 7. n n F h,h n F h,h n 5.8 The proof follows from the latter theorem. 6 Generatng Functon for h-fbonacc Operators In ths secton, the generatng functons for the h-fbonacc operators are gven. As a result, the h-fbonacc operator sequences are seen as the coeffcents of the power seres of the correspondng generatng functon. To derve a generatng functon for h-fbonacc operators, consder the functon gx gven by : gx F x 6. It follows that gxf 0 x 0 F x gxx 6

17 Hence gxx 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 gxλ + λ x gx x+λ+ λ x gx x x x+λ + λ x x λ + xλ x So the generatng functon for the h-fbonacc operators s : gx x λ + xλ x F x 6. Next we lst the generatng functons that generate the varous powers and products of the h-fbonacc sequences. Theorem 8. We have x λ + x λ x F x 6.3 Usng the h-bnet formula, we have : F x λ + λ x λ + x λ x λ + x λ x λ + λ x λ + xλ x x λ + xλ x 7

18 Theorem 9. We have Usng the h-bnet formula, we have : F + x +λ + λ x λ + x λ x F + x 6.4 λ+ + λ + x λ + + x λ+ λ + x λ λ x +λ + λ x λ + xλ x +λ + λ x λ + xλ x λ + x Theorem 0. We have F m +λ + λ F m x λ + xλ x Usng the h-bnet formula, we have : F m+n x n n0 n0 F m+n x n 6.5 λ m+n + λ m+n x n n0 λ m+n + xn n0 λ m+n x n λ m + λ + x λm λ x λm + λm +λ +λ λ m + λ m x λ + xλ x F m λ + λ F m x λ + xλ 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 6.6 F F + x 6.7 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 usng h-bnet s formula. The followng proposton gves us the value for the h-fbonacc sequence seres wth weghts p +. Theorem. For each non-vanshng nteger number p : 0 0 F p + Usng the h-bnet s formula, we get : F + p + λ p +.λ 0 p p p+λ + λ pλ + pλ λ+ p 0 λ p Equaton 6.0 yelds the followng results for partcular values of p : When p When p 3 When p 8 When 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 n appearng n the denomnator s the usual Fbonacc number. 9

20 7 Conclusons h-analogue of Fbonacc numbers have been ntroduced and studed. Several propertes of these numbers are derved. In addton, the h-bnet s formula for these numbers s found and the generatng functon of these h-fbonacc sequences and ther varous powers have been deduced. It s straghtforward to ntroduce the h-lucas numbers. Ths wor s n progress [4. It s possble to ntroduce the q h-analogue of Fnonacc numbers by usng the q h-analogue of bnomal coeffcents whch was ntroduced n [5. Indeed the q h-analogue of bnomal coeffcents was found to be : [ [ [ n n h h n ;[ hh 7. h;[ q,h,h These coeffcents obey to the followng propertes : q q and [ n+ q,h,h [ n+ + q [ n q,h,h q,h,h hh [n+ q [ + q +hh [ n [ n q,h,h + q,h,h + The q h-analogue of Fbonacc numbers wll be defned as follows : q F h,h n+ [ n q [ n and they obey the followng recurrence formula : q,h,h q F h,h n+ q F h,h n +q n q F h,h + n 7.5 For hh and h 0, the q h-analogue of Fbonacc numbers are just the q-fbonacc numbers see [6. Smlarly, several propertes of the q h-analogue of Fbonacc numbers can be derved. We lst below some of them whch are easy to prove from the recurrence formula. hh q q F h,h + q F h,h n+ q h n h q ;n F h,h +n q F h,h n h n h ;n q F h,h +n /q q F h,h n+ hn h ;n 7.6 0

21 Acnowledgments I would le to than Tom Koornwnder for hs helpful comments and suggestons. References [ T. Koshy, Fbonacc and Lucas Numbers wth Applcatons, Wley, New Yor, 00. [ H. Benaoum, h analog of Newton s bnomal formula, J.Phys.A:Math. Gen. 3 L75,998. e-prnt: math-ph/980. [3 T.S. Chhara, An Introducton to Orthogonal Polynomals, Gordon and Breach, New Yor, 978. [4 H. Benaoum, h-analogue of Lucas numbers, wor n progress. [5 H. Benaoum, q,h - analog of Newton s bnomal formula, J.Phys.A:Math. Gen ,999. e-prnt: math-ph/9808. [6 L. Carltz, Fbonacc notes. 3 : q-fbonacc numbers, Fbonacc Quarterly ; Fbonacc notes 4 : q-fbonacc polynomals, Fbonacc Quarterly 3 97, 975.