A GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by

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1 A GENERALIZATION OF A CONJECTURE OF MELHAM EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 Abstact A genealization of one of Melha s conectues is pesented Afte witing it in tes of Gaussian binoial coefficients, a solution is found using the eleentay techniue of patial faction decoposition 1 Intoduction The Fibonoial coefficient is, fo n 1, defined by { } n F 1 F F n : F 1 F F n F 1 F F n n with 1 whee F n is the nth Fibonacci nube n 0 Fo a detailed discussion of the Fibonoial coefficients, we efe to the list of efeences in [] [ ] n The Gaussian binoial coefficient is defined, fo all eal n and integes with 0, by [ ] n ; n : ; ; n and as zeo othewise, whee a; n 1 a1 a 1 a n 1 [ ] n Thus, is a ational function of the paaete Fo oe details, see [1] Let α and β 1 5 Then the well known Binet fos give and thus F n αn β n α β, L n α n + β n, n 1 1 n F n α 1, L n α n 1 + n, with β/α α, so that i α, whee F n is the nth Fibonacci nube and L n is the nth Lucas nube All the identities we ae going to deive hold fo 000 Matheatics Subect Classification 11B37 Key wods and phases Melha s Conectue, Fibonoial coefficients 1

2 EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 geneal, and esults about Fibonacci and Lucas nubes coe out as coollaies fo the special choice of The link between the Fibonoial and Gaussian binoial coefficients is { } [ ] n n α n with α Melha [3] deived soe failies of identities between sus of powes of the Fibonacci and Lucas nubes He also conectued a coplex identity by using a falling Fibonacci factoial F n, which begins at F n fo n 0, and is the poduct of Fibonacci nubes excluding F 0 His conectue is in two pats: Let, n, k Z with 1 Then: a F +1 n+k+ + 1 F 1 F +1k+ b The Lucas countepat of a, L +1 n+k+ + 1 F 1 F +1k+ F +1 n k F +1k+ 1 L +1 n k F +1k+ 1 F +1n+ {5 +1 F +1n+ if is odd, 5 L +1n+ if is even In [], we eaanged the conectues a and b by using Fibonoials and then gave a solution of the conectue by tanslating it into a expession: we wee left with the evaluation of a cetain su This was achieved using contou integation The pesent pape is oganized as follows: i We genealize the conectue of Melha by using indices in aithetic pogession fo both, the Fibonacci and the Lucas instance ii Then we give a solution fo this geneal foula by a patial faction decoposition ethod that is even siple than the contou integation given in [] although it is essentially euivalent A genealization of Melha s Conectue In this section, we give a genealization of the Fibonoial coefficients in ode to state a geneal vesion of the conectue of Melha Definition 1 Fo all integes n,, with, 1, the Fibonoial is defined as { } n : F nf n F n F F F with { } n 0 1 whee F n is the nth Fibonacci nube Now we give a genealization of the conectue of Melha in tes of Fibonoials: Theoe 1 Fo any integes, n and k:

3 A GENERALIZATION OF A CONJECTURE OF MELHAM 3 i If odd, then 1 1 ii If even, then + 1k + + 1k F +1 n k k + + 1k iii The Lucas countepat of i, 1 1 L +1 n+k+ + 1 F +1 n k 1 F +1 n+k+ F +1k+ F +1n+ F +1 n+k+ F +1k+ F +1n+ + 1k + + 1k L iv The Lucas countepat of ii, n k 1 F +1k+ F +1n+ if is odd, F +1k+ L +1n+ 1 if is even + 1k + + 1k L +1 n k 1 5 L +1 n+k+ F +1k+ F +1n+ if is odd, F +1k+ L +1n+ 1 if is even Poof We want to point out that ewiting i iv in tes of binoials poduces the sae expession fo i and ii and also fo iii and iv We can cobine all the cases i iv in one foula as follows: 1 +1k+ +1k+ ; ; [ ] h n+k k+ 1 1 h n+ +1k+ ; 1 +1k h n k +1 1

4 4 EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 Hee, h 1 in 1 gives us the notation of the cases i, ii and siilaly h 0 in 1 gives the cases iii and iv Fo the poof, let us conside h z n+k+ +1 X : 1 z1 z 1 z z1 z +1k+ Pefoing patial faction decoposition, we get: with X ; ; C 1 z +1k+ + 1 z h z n+k+ +1 C z1 z1 z 1 z h n+k k+ 1 1 z z +1k We note that the degee of nueato and denoinato is + 1 As z, X B z + Oz with Futheoe, B 1 1 h n+k k+ 1 h+1+1 n+1++1 zx and letting z appoach : B Thus, we get: 1 +1 ; ; ; ; h n+k k+ z 1 z h n+k k+ 1 h 1+1 n+1++1 ; [ ] h n+k k+ Cz + 1, 1 z +1k+ C k+ C ; +1k+ + ;,

5 A GENERALIZATION OF A CONJECTURE OF MELHAM 5 o [ ] h n+k k+ C ; +1k+ + ; 1 h 1+1 n+1++1 ; Now we wok out the constant C: 1 C +1k h n k k 1 +1k 1 +1k h n k +1 +1k+ +1k 1 +1k+ 1 +1k+ 1 +1k h n k +1 +1k k+ 1 +1k+ 1 +1k+ k h n k k+ ; Using and 3, we obtain [ ] h n+k k h n k +1 +1k++1 +1k+ ; ; + ; 1 h 1+1 n+1++1 ; Now we show that this is euivalent to 1 We wite 1, but eplace the su by the foula that we ust obtained and get 1 +1k+ +1k+ ; ; [ h n k +1 +1k k+ ; ; ] + ; 1 h 1+1 n+1++1 ; 1 1 h n+ +1k+ ; 1 +1k h n k +1 Ou goal is achieved once we see that this is an identity Fo that, we will gadually siplify it until a tivial identity eains

6 6 EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 We need to pove that 1 +1k+ +1k+ ; [ h n k +1 +1k k+ ; ] h 1+1 n h n+ +1k+ ; 1 +1k+ Cobining the fist two factos, h n k +1 +1k+ ; [ h n k +1 +1k k+ ; ] h 1+1 n h n+ +1k+ ; 1 +1k+ which we futhe ewite: h n k +1, h n k +1 +1k k+ ; 1 h 1+1 n k+ ; 1 1 h n+ +1k+ ; o o k h n k +1, +1k+ ; 1 h+1+1 n k+ ; 1 1 h n+ +1k+ ;, 1 h 1+1 n k+ ; 1 h n+ +1k+ ; Finally, we see that this is coect, and so we have poved the identities i iv Refeences [1] G E Andews, The theoy of patitions, Encyclopedia of atheatics and its applications, Vol, Addison-Wesley, Reading, MA, 1976 [] E Kılıç, I Akkus, and H Podinge, A poof of a conectue of Melha, The Fibonacci Quately, accepted

7 A GENERALIZATION OF A CONJECTURE OF MELHAM 7 [3] R S Melha, Failies of identities involving sus of powes of the Fibonacci and Lucas nubes, The Fibonacci Quately , TOBB Univesity of Econoics and Technology Matheatics Depatent Sögütözü Ankaa Tukey E-ail addess: ekilic@etuedut 1 Ankaa Univesity Matheatics Depatent Ankaa Tukey E-ail addess: iakkus@scienceankaaedut 3 Depatent of Matheatics, Univesity of Stellenbosch 760 Stellenbosch South Afica E-ail addess: hpoding@sunacza 3