A PROOF OF A CONJECTRE OF MELHAM EMRAH KILIC, ILKER AKKS, AND HELMT PRODINGER Abstract. In this paper, we consider Melha s conecture involving Fibonacci and Lucas nubers. After rewriting it in ters of Fibonoial coefficients, we give a solution of the conecture by evaluating a certain -su using contour integration. 1. Introduction The Fibonacci nubers are defined for n > 0 by F n+1 F n +F n 1 where F 0 0 and F 1 1. The Lucas nubers are defined for n > 0 by L n+1 L n +L n 1 where L 0 and L 1 1. The Binet fors of the Fibonacci and Lucas seuences are F n αn β n and L n α n +β n α β where α, β are (1± 5/. Recently as an interesting generalization of the binoial coefficients, the Fibonoial coefficients have taken the interest of several authors. For their properties, we refer to [1, 6, 8, 9, 11, 14, 15, 17, 19, 1, ]. The Fibonoial coefficient is defined, for n 1, by { n F 1 F F n (F 1 F F n (F 1 F F F n! F n!f! with { n { n n 0 1 where Fn is the nth Fibonacci nuber and F n! F 1 F F n is nth Fibonacci factorial. These coefficients satisfy the relation: { n F +1 { n 1 { n 1 +F n. (1.1 In [8], Hoggatt considered Fibonoial coefficients with indices in arithetic progressions. For exaple, he defined the following generalization by taking F kn instead of F n for a fixed positive integer k: { n k F k F k F kn ( Fk F k F k(n ( Fk F k F k. AGST 010 41
THE FIBONACCI QARTERLY The Fibonoials appear in several places in the literature; we give two exaples: the n n right-adusted Pascal atrix P n whose (i, entry is given by ( 1 (P n i. +i n 1 Carlitz [] was the first to link the Fibonoial coefficients with the auxiliary polynoial of the right-adusted Pascal atrix P n via its coefficients. Sincethen, soe relationships between the generalized Pascal atrix and the Fibonoial coefficients have been constructed, and the Fibonoial coefficients have been studied by soe authors [, 3, 1, 13, 0]. Secondly, in [5, 7, 8, 10], one can find that the nth powers of Fibonacci nubers satisfy the following auxiliary polynoial n { C n (x ( 1 i(i+1/ n x n i, (1. i i0 where { n i is defined as before. Torreto and Fuchs [1] considered the general second order recurrence relation y n+ gy n+1 hy n, h 0. (1.3 Let α and β be the roots of the auxiliary polynoial f(x x gx+h of (1.3. Let n and V n be two solutions of (1.3 defined by n { α n β n α β if α β nα n 1 if α β and V n α n +β n. In [10, pp. 4 45], Jarden showed that the kth order recurrence relation k { k ( 1 z n+k 0 0 h (+1 holds for the product z n of the nth ters of k 1 seuences satisfying (1.3, where { k { k k 1 k +1 k, 1. 1 0 In [1], the authors established soe identities involving the { k, one of which is the forula: k { k ( 1 0 h (+1 a1 +k a +k ak +k y n+k 1 k y n+a1 + +a k +k(k+1/, where y n and n satisfy (1.3 and n and the a s are any integers. Melha [16] derived failies of identities between sus of powers of the Fibonacci and Lucas nubers. In his work, while deriving these identities, he conectured a coplex identity between the Fibonacci and Lucas nubers. We recall this conecture: (a Let k,,n Z with > 0, show that 0 F +1 n+k+ +( 1 (+3 (F 1 ( F (+1k+ F +1 n k F (+1(n+. (1.4 F (+1k+ 4 VOLME 48, NMBER 3 1
(b The Lucas counterpart of (a is given by 0 L +1 n+k+ +( 1 (+3 (F 1 ( F (+1k+ A PROOF OF A CONJECTRE OF MELHAM L +1 n k F (+1k+ 1 { 5 +1 F (+1(n+ if is odd, 5 L (+1(n+ if is even, where (F n ( is the falling factorial, which begins at F n for n 0, and is the product of Fibonacci nubers excluding F 0. For exaple (F 6 (5 F 6 F 5 F 4 F 3 F and (F 3 (5 F 3 F F 1 F 1 F. For > 0, define (F 0 ( F 1 F F and (F 0 (0 1. In this paper, we first rearrange the conecture of Melha by using Fibonoial coefficients instead of the falling Fibonacci factorial. After this, we give a solution of the conecture by translating it into a -expression; we are left with the evaluation of a certain su. This is achieved using contour integration.. The Conecture in Ters of Fibonoials In this section, before solving the conecture, we will need to rewrite it via the Fibonoials. One can obtain the following version of the conecture (a by considering the definitions of the falling Fibonacci factorial and the Fibonoials coefficients; the following euation is euivalent to (1.4. Notice that 0 F +1 n+k+ (F 1 ( 1 (F 0 F (+1k+ (F (+1k+ (+1k+ (F (+1k (+1k +( 1 (+3 F +1 n k ( 1 F (+1k+ F (+1(n+. (F 0 (F ( 1 [# of even positive integers ] (F ( 1 (+3. Fro this the following conecture is straight-forward. Conecture 1. For any integers,n and k, ( 1 (+3 0 +( 1 (+3 F +1 { { (+1k + (+1k + 1 n k ( 1 1 F (+1k+ F (+1(n+. F +1 n+k+ AGST 010 43
THE FIBONACCI QARTERLY Proof. By taking β/α, the claied euality is reduced to the following for: [ ] (+1k + (1 (+1k+ [ ] ( 1 (+1/ 0 (1 n+k+ +1 1 (+1k+ (1 (+1(n+ (;(+1k+ ( 1 (+1(k+1 (1 n k +1 (.1 (; (+1k where [ ] stands for the Gaussian -binoial coefficient: [ ] (; : (; (; with Define (z; n : (1 z(1 z (1 z n 1. [ ] S : ( 1 (+1/ 1 By contour integration (for siilar exaples, see [18], I R 1 (; πi (z; 1 z (1 n+k+ +1 1 (+1k+. (1 z n+k+ +1 dz. 1 z (+1k+ The integration is over a large circle with radius R. We evaluate the integral by residues. The poles at 1,..., ( lead to our su, but there are other poles: at z 0, at z 1, and at z (+1k. Then we get (; 1 I R S +Res z0 (z; z (; 1 +Res z1 (z; z (; 1 +Res z (+1k (z; z (1 z n+k+ +1 1 z (+1k+ (1 z n+k+ +1 1 z (+1k+ (1 z n+k+ +1 1 z (+1k+. For the reader s convenience, we copute one of these residues in a separate coputation: (; 1 Res z1 (z; z (1 z n+k+ +1 1 z (+1k+ li(z 1 (; 1 z 1 (z; z (; 1 (z; z (1 n+k+ +1 1 (+1k+. (1 z n+k+ +1 1 z (+1k+ (1 z n+k+ +1 1 z (+1k+ 44 VOLME 48, NMBER 3 z1
A PROOF OF A CONJECTRE OF MELHAM Note that as z gets large, (; 1 (z; z Conseuently, as R, (1 z n+k+ +1 1 z (+1k+ 1 (; (n+k+(+1 z ( (+1k+ 1 z (; n(+1+(+1. Thus, I R (; n(+1+(+1. (; n(+1+(+1 S +(; (1 n+k+ +1 1 (+1k+ +[(z (+1k 1 ] (; 1 (z; z So we obtain S (; n(+1+(+1 +(; (1 n+k+ +1 (; ( (+1k ; (1 n k +1 1 (+1k+ (; n(+1+(+1 +(; (1 n+k+ +1 1 (+1k+ ( 1 (; (; (+1k (+1k+ (+1 (1 n k +1. (; (+1k+ Now we ust prove that [ ] (+1k + (1 (+1k+ (1 n+k+ +1 1 (+1k+ [ ] (+1k + (1 (+1k+ (; n(+1+(+1 [ ] (+1k + +(1 (+1k+ (; [ ] (+1k + (1 (+1k+ (1 n+k+ +1 1 (+1k+ [ ] (+1k + (1 (+1k+ (1 z n+k+ +1 1 z (+1k+. ( 1 (; (; (+1k (+1k+ (+1 (1 n k +1 (; (+1k+ (1 (+1(n+ (;(+1k+ ( 1 (+1(k+1 (1 n k +1. (; (+1k AGST 010 45
THE FIBONACCI QARTERLY We gradually siplify the euation that ust be proved: [ ] (+1k + (1 (+1k+ (; n(+1+(+1 [ ] (+1k + +(1 (+1k+ (; [ ] (+1k + (1 (+1k+ ( 1 (; (; (+1k (+1k+ (+1 (1 n k +1 (; (+1k+ or or (1 (+1(n+ (; (+1k+ (; (+1k ( 1 (+1(k+1 (1 n k +1, [ ] (+1k + (1 (+1k+ [ ] (+1k + +(1 (+1k+ (1 (+1(n+ (; (+1k+ (; (+1k, [ ] (+1k + (1 (+1k+ (1 (+1(n+ (;(+1k+. (; (+1k (; n(+1+(+1 (; ( (; 1 n(+1+(+1 Conseuently we are left to prove that [ ] (+1k + (1 (+1k+ (; (; (+1k+. (; (+1k Since this is true, the proof is coplete. Note that this proof establishes (.1 for all values of, not ust for β/α. Siilarly the Lucas counterpart of the conecture is rewritten in ters of the Fibonoials as follows: for odd, 0 ( 1 (+3 { { (+1k + (+1k + 1 +( 1 (+3 L +1 n k 5+1 ( 1 1 L +1 n+k+ F (+1k+ F (+1(n+, 46 VOLME 48, NMBER 3
and for even, 0 ( 1 (+3 A PROOF OF A CONJECTRE OF MELHAM { { (+1k + (+1k + 1 +( 1 (+3 L +1 n k 5 ( 1 1 F (+1k+ L (+1(n+. L +1 n+k+ By taking β/α, the above eualities are translated to the following fors: [ ] (+1k + [ ] (1 (+1k+ ( 1 (+1/ (1+ n+k+ +1 1 (+1k+ (. 0 (1 (+1(n+ (;(+1k+ ( 1 (+1(k+1 (1+ n k +1 (; (+1k and [ ] (+1k + (1 (+1k+ 0 ( 1 (+1/ [ (1 (+1(n+ (;(+1k+ ( 1 (+1(k+1 (1+ n k +1, (; (+1k ] (1+ n+k+ +1 1 (+1k+ (.3 respectively. Note that (. and (.3 are the sae; it is no ore necessary to distinguish the parity of. The proof of the euality (. (and (.3 can be done siilarly to (.1. Again, it holds for general. 3. Acknowledgent The constructive rearks of one referee are gratefully acknowledged. References [1] J. E. Andersen and C. Berg, Quantu Hilbert atrices and orthogonal polynoials, J. Coput. Appl. Math, 33 (009, 73 79. [] L. Carlitz, The characteristic polynoial of a certain atrix of binoial coefficients, The Fibonacci Quarterly, 3.1 (1965, 81 89. [3] C. Cooper and R. Kennedy, Proof of a result by Jarden by generalizing a proof by Carlitz, The Fibonacci Quarterly, 33.4 (1995, 304 310. [4] L. A. G. Dresel, Transforations of Fibonacci-Lucas identities, Applications of Fibonacci Nubers, 5. (1993, 169 184. [5] P. Duvall and T. Vaughan, Pell polynoials and a conecture of Mahon and Horada, The Fibonacci Quarterly, 6.4 (1988, 344 353. [6] H. W. Gould, The bracket function and Fontené Ward generalized binoial coefficients with application to Fibonoial coefficients, The Fibonacci Quarterly, 7.1 (1969, 3 40. [7] A. P. Hillan and V. E. Hoggatt, The characteristic polynoial of the generalized shift atrix, The Fibonacci Quarterly, 3. (1965, 91 94. [8] V. E. Hoggatt Jr., Fibonacci nubers and generalized binoial coefficients, The Fibonacci Quarterly, 5.4 (1967, 383 400. [9] A. F. Horada, Generating functions for powers of a certain generalized seuence of nubers, Duke Math. J., 3 (1965, 437 446. [10] D. Jarden, Recurring Seuences, Riveon Leateatika, Jerusale, Israel, 1958. AGST 010 47
THE FIBONACCI QARTERLY [11] D. Jarden and T. Motzkin, The product of seuences with a coon linear recursion forula of order, Riveon Leateatika, 3 (1949, 5 7, 38. [1] E. Kilic, The generalized Fibonoial atrix, European J. Cob., 31 (010, 193 09. [13] E. Kilic, G. N. Stanica and P. Stanica, Spectral Properties of Soe Cobinatorial Matrices, 13th International Conference on Fibonacci Nubers and Their Applications, 008. [14] D. A. Lind, A deterinant involving generalized Binoial coefficients, The Fibonacci Quarterly, 9.(1971, 113 119, 16. [15] C. T. Long, Discovering Fibonacci identities, The Fibonacci Quarterly, 4. (1986, 160 167. [16] R. S. Melha, Failies of identities involving sus of powers of the Fibonacci and Lucas nubers, The Fibonacci Quarterly, 37.4 (1999, 315 319. [17] H. Prodinger, On a uestion of Cooper and Kennedy, The Fibonacci Quarterly, 35. (1997, 135 136. [18] H. Prodinger, Soe applications of the -Rice forula, Rando Structures and Algoriths, 19.3-4 (001, 55 557. [19] J. Seibert and P. Troovsky, On soe identities for the Fibonoial coefficients, Math. Slovaca, 55 (005, 9 19. [0] P. Stanica, Netted atrices, Int. J. Math. Math. Sci., 39 (003, 507 518. [1] R. F. Torretto and J. A. Fuchs, Generalized binoial coefficients, The Fibonacci Quarterly,.4 (1964, 96 30. [] P. Troovsky, On soe identities for the Fibonoial coefficients via generating function, Discrete Appl. Math., 155.15 (007, 017 04. MSC010: 11B37 TOBB niversity of Econoics and Technology, Matheatics Departent 06560, Sögütözü, Ankara, Turkey E-ail address: ekilic@etu.edu.tr Ankara niversity, Matheatics Departent 06100, Ankara, Turkey E-ail address: iakkus@science.ankara.edu.tr Departent of Matheatics, niversity of Stellenbosch 760, Stellenbosch, South Africa E-ail address: hproding@sun.ac.za 48 VOLME 48, NMBER 3