A PROOF OF MELHAM S CONJECTURE
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1 A PROOF OF MELHAM S CONJECTRE EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 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 α β and L n α n + β n where α, β are (1 ± 5 )/. Recently as an interesting generalization of the binoial coefficients, the Fibonoial coefficients have been taken 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 by the relation for n 1 { 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 + F n { n 1. (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: 000 Matheatics Subect Classification. 11B37. Key words and phrases. Melha s Conecture, Fibonoial coefficients, -binoial coefficients, contour integration. 1
2 EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 { n F k F k... F ( )( kn ). k Fk F k... F k(n ) Fk F k... F k 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 [] firstly showed that the Fibonoial coefficients appear in the auxiliary polynoial of the right-adusted Pascal atrix P n as the coefficients of it. Since soe relationships between the generalized Pascal atrix and the Fibonoial coefficients have been constructed, 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 nα n 1 if α β if α β and V n α n + β n. In [10], Jarden showed that the kth order recurrence relation k { k ( 1) z n+k 0 h (i+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, In [1], the authors established soe identities involving the { k is the forula: k { k ( 1) h (i+1) a1+k a+k... ak +k y n+k, one of which 1... k y n+a1+ +a k +k(k+1)/, where y n and n satisfying (1.3) and n and the a s are any integers. Melha [16] derived the 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. Now we recall this conecture:
3 A PROOF OF MELHAM S CONJECTRE 3 (a) Let k,, n Z with > 0, show that F +1 n+k+ (F 1 ) () F (+1)k+ (b) The Lucas counterpart of (a) is given by L +1 n+k+ (F 1 ) () F (+1)k+ F +1 n k F (+1)k+ 1 F (+1)(n+ ). (1.4) L +1 n k F (+1)k+ 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, first we 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: Conecture 1. For any integers, n and k, ( 1) (+3) { { ( + 1)k + ( + 1)k + 1 F +1 1 n k ( 1 F +1 n+k+ F (+1)k+ )F (+1)(n+ ). Proof. By taking β/α, the claied euality is reduced to the following for: ( + 1)k + (1 (+1)k+ ) ( 1) (+1)/ (1 n+k+ ) +1 1 (+1)k+ (1 (+1)(n+) ) (; )(+1)k+ (; ) (+1)k ( 1) (k+1) (1 n k ) +1 (.1)
4 4 EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 where stands for the Gaussian -binoial coefficient: with Define : (; ) (; ) (; ) (z; ) n : (1 z)(1 z)... (1 z n 1 ). S : ( 1) (+1)/ 1 (1 n+k+ ) +1 1 (+1)k+. By contour integration (for siilar exaples, see [18]), I R 1 (; ) πi (z; ) 1 z (1 z n+k+ ) +1 dz. 1 z (+1)k+ 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 (+1)k. Then we get (; ) 1 I R S + Res z0 (z; ) z (; ) 1 + Res z1 (z; ) z (1 z n+k+ ) +1 1 z (+1)k+ (1 z n+k+ ) +1 1 z (+1)k+ (; ) 1 + Res z (+1)k (z; ) z Note that as z gets large, (; ) 1 (z; ) z Conseuently, as R, Thus (1 z n+k+ ) +1 1 z (+1)k+. (1 z n+k+ ) +1 1 z 1 (; ) (n+k+)(+1) (+1)k+ z ( ) (+1)k+ n(+1)+ I R (; ). 1 z (; ) n(+1)+. n(+1)+ (; ) S + (; ) (1 n+k+ ) +1 1 (+1)k+ + [(z (+1)k ) 1 ] (; ) 1 (z; ) z (1 z n+k+ ) +1 1 z (+1)k+.
5 A PROOF OF MELHAM S CONJECTRE 5 So we obtain n(+1)+ S (; ) + (; ) (1 n+k+ ) +1 1 (+1)k+ (; ) (1 n k ) +1 ( (+1)k ; ) n(+1)+ (; ) + (; ) (1 n+k+ ) +1 1 (+1)k+ ( 1) (; ) (; ) (+1)k k+ (1 n k ) +1. (; ) (+1)k+ Now we ust prove that ( + 1)k + (1 (+1)k+ (1 n+k+ ) +1 ) 1 (+1)k+ ( + 1)k + (1 (+1)k+ ) (; ) ( + 1)k + + (1 (+1)k+ ) (; ) ( + 1)k + (1 (+1)k+ (1 n+k+ ) +1 ) 1 (+1)k+ ( + 1)k + (1 (+1)k+ ) n(+1)+ ( 1) (; ) (; ) (+1)k k+ (1 n k ) +1 (; ) (+1)k+ ) (1 (+1)(n+) (; )(+1)k+ ( 1) (k+1) (1 n k ) +1. (; ) (+1)k We gradually siplify the euation that ust be proved: ( + 1)k + (1 (+1)k+ ) (; ) ( + 1)k + + (1 (+1)k+ ) (; ) ( + 1)k + (1 (+1)k+ ) n(+1)+ ( 1) (; ) (; ) (+1)k k+ (1 n k ) +1 (; ) (+1)k+ (1 (+1)(n+) ) (; ) (+1)k+ ( 1) (k+1) (1 n k ) +1, (; ) (+1)k
6 6 EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 or or ( + 1)k + (1 (+1)k+ ) (; ) ( + 1)k + + (1 (+1)k+ ) (; ) (1 (+1)(n+) ) (; ) (+1)k+ (; ) (+1)k, ( + 1)k + (1 (+1)k+ ) ( (; ) 1 n(+1)+ n(+1)+ ) (1 (+1)(n+) (; )(+1)k+. (; ) (+1)k Conseuently we are left to prove that ( + 1)k + (1 (+1)k+ ) (; ) (; ) (+1)k+. (; ) (+1)k 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, ( 1) (+3) and for even, ( 1) (+3) { { ( + 1)k + ( + 1)k ( L +1 n k { { ( + 1)k + ( + 1)k ( L +1 n k 5 1 L +1 n+k+ ) F (+1)k+ )F (+1)(n+ ), L +1 n+k+ F (+1)k+ )L (+1)(n+ ). By taking β/α, the above eualities are translated to the following fors: ( + 1)k + (1 (+1)k+ ) ( 1) (+1)/ (1 + n+k+ ) +1 (.) 1 (+1)k+ (1 (+1)(n+) ) (; )(+1)k+ (; ) (+1)k ( 1) (k+1) (1 + n k ) +1
7 A PROOF OF MELHAM S CONJECTRE 7 and ( + 1)k + (1 (+1)k+ ) ( 1) (+1)/ (1 + (+1)(n+) (1 + n+k+ ) +1 1 (+1)k+ (.3) ) (; )(+1)k+ (; ) (+1)k ( 1) (k+1) (1 + n k ) +1 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. References [1] J. E. Andersen and C. Berg, Quantu Hilbert atrices and orthogonal polynoials, J. Coput. Appl. Math, in press. [] L. Carlitz, The characteristic polynoial of a certain atrix of binoial coefficients, Fibonacci Quart. 3 (1965), [3] C. Cooper and R. Kennedy, Proof of a result by Jarden by generalizing a proof by Carlitz, Fibonacci Quart. 33 (4) (1995), [4] L. A. G. Dresel, Transforations of Fibonacci-Lucas identities, In Applications of Fibonacci Nubers, 5 (1993), [5] P. Duvall and T. Vaughan, Pell polynoials and a conecture of Mahon and Horada, Fibonacci Quart. 6 (4) (1988), [6] H. W. Gould, The bracket function and Fountené Ward generalized binoial coefficients with application to Fibonoial coefficients, Fibonacci Quart. 7 (1969), [7] A. P. Hillan and V. E. Hoggatt, The characteristic polynoial of the generalized shift atrix, Fibonacci Quart. 3 () (1965), [8] V. E. Hoggatt Jr., Fibonacci nubers and generalized binoial coefficients, Fibonacci Quart. 5 (1967), [9] A. F. Horada, Generating functions for powers of a certain generalized seuence of nubers, Duke Math. J. 3 (1965), [10] D. Jarden, Recurring seuences, Riveon Leateatika, Jerusale, Israel, 1958, [11] D. Jarden, 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., in press. [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, Fibonacci Quart. 9 () (1971) , 16. [15] C. T. Long, Discovering Fibonacci identities, Fibonacci Quarterly, 4 () (1986), [16] R. S. Melha, Failies of identities involving sus of powers of the Fibonacci and Lucas nubers, Fibonacci Quarterly, 37 (4) (1999), [17] H. Prodinger, On a uestion of Cooper and Kennedy. H. Prodinger, Fibonacci Quarterly 35 (1997) [18] H. Prodinger, Soe applications of the -Rice forula, Rando Structures and Algoriths 19 (3-4) (001), [19] J. Seibert and P. Troovsky, On soe identities for the Fibonoial coefficients, Math. Slovaca 55 (005), [0] P. Stanica, Netted atrices, Int. J. Math. Math. Sci. 39 (003), [1] R. F. Torretto and J. A. Fuchs, Generalized Binoial coefficients, Fibonacci Quart. (1964), [] P. Troovsky, On soe identities for the Fibonoial coefficients via generating function. Discrete Appl. Math. 155 (15) (007),
8 8 EMRAH KILIC 1, ILKER AKKS, AND HELMT PRODINGER 3 1 TOBB niversity of Econoics and Technology Matheatics Departent Sögütözü Ankara Turkey E-ail address: ekilic@etu.edu.tr 1 Ankara niversity Matheatics Departent Ankara Turkey E-ail address: iakkus@science.ankara.edu.tr 3 Departent of Matheatics, niversity of Stellenbosch 760 Stellenbosch South Africa E-ail address: hproding@sun.ac.za 3
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