EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS

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1 EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal weight fuctio We use the patial factio decompositio techiue to pove the claimed esults We also give some iteestig applicatios of ou esults to cetai geealized Fiboomial sums weighted with fiite poducts of ecipocal Fiboacci o Lucas umbes Itoductio Defie the secod ode liea seueces {U ad {V fo 2 by U = pu + U 2, U 0 = 0, U =, V = pv + V 2, V 0 = 2, V = p Fo ad a itege m, defie the geealized Fiboomial coefficiet with idices i a aithmetic pogessio by { U m U 2m U m := U;m U m U 2m U m U m U 2m U m with { 0 U;m = { = Whe p = m =, we obtai the usual Fiboomial coefficiets, U;m deoted by { Whe m =, we obtai the geealized Fiboomial coefficiets, deoted F by { U Thoughout this pape we will use the followig otatios: the -Pochhamme symbol x; = x x x ad the Gaussia -biomial coefficiets [ ] ; = ; ; I this pape, we fistly coside a sum of the Gaussia -biomial coefficiets with a paametic atioal weight fuctio of the fom: fo ay positive itege w, ay ozeo eal umbe a, oegative itege, iteges t ad such that [ ] + 2 t a ; w + We will compute this sum by cosideig a appopiate patial factio decompositio 2000 Mathematics Subect Classificatio B65, 05A30, B39 Key wods ad phases -Biomial coefficiet, Fiboomial coefficiets, patial factio decompositio, idetities

2 Recetly, the authos of [2, 3] computed cetai Fiboomial sums with geealized Fiboacci ad Lucas umbes as coefficiets Fo example, if ad m ae both oegative iteges, the whee { 2 U 2m = P,m { 2 + { 2 { 2 + U 2m = P,m V 2m = P,m V 2m = P,m m = m m = m m V 2 if m, P,m = m V 2 if < m; = { 2m U 4 2, 2 { 2m 2 { 2m 2 { 2m 2 U 2+2, V 4 2, V 2+2, alteatig aalogues of these sums wee also evaluated We will peset hee some iteestig applicatios of ou esults to sums of Fiboomial coefficiets with atioal weight fuctios These ids of Fiboomial sums ivolvig atioal weight fuctios have ot bee cosideed befoe i the liteatue, to the best of ou owledge Ou appoach egadig these applicatios is as follows We use the Biet foms U = α β α β = α ad V = α + β = α + with = β/α = α 2, so that α = i/ whee α, β = p ± /2 ad = p The li betwee the geealized Fiboomial ad Gaussia -biomial coefficiets is { = α m [ ] with = α 2 U;m m Thus, the evaluatios ca be doe i the -wold ad the be etaslated ito the laguage of Fiboomial coefficiets ad the lie 2 The mai esults We give ou mai esult o computig Gaussia -biomial sums with a paametic atioal weight fuctio: Theoem Fo ay positive itege w, ay ozeo eal umbe a, oegative itege, iteges t ad such that, [ ] + 2 +t a ; w + 2

3 = a t w+ ; w ; w w ; w a w ; + t + t + + w+ 2 + tw a w 2 tw Poof We coside the patial factio decompositio of the fuctio hz = z z z z t+ z + z az a w z a w The patial factio decompositio of hz taes the fom: + 2 +t+ hz = a ; w + z ; ; + F 0, t, a, z a + G, t, a, + F, t, a, + + F, t, a, z a w z a w + + G t, t, a, z z t Now we multiply this elatio by z ad let z ad obtai 0 = lim z + 2 +t+ z z a ; w + ; ; z + F 0, t, a, z a + F z, t, a, z a w z + + F, t, a, z a + G, t, a,, w which gives us the euatio + 2 +t+ 0 = + a ; w + ; ; o whee ad + 2 +t a ; w + ; ; = F, t, a, + G, t, a, F, t, a, + G, t, a, F, t, a, = z z z z t+ G, t, a, = [z ]hz z + z=a z a z a w z a +w z a w w 3

4 Fist we wo out F, t, a, : a t w t F, t, a, = a w a w+ a w+ a w aa w a w a w a w a w a +w a w a +2w a w a w = a t w t a w ; + Now we compute G, t, a, : a 2 w w w w a w w 2w w = a t w t 2 w w a w ; + w ; w w ; w = a t wt+ + 2 w a w ; + w ; w w ; w G, t, a, = [z ]hz = [z +t ] z z z z az a w z a w = [z +t ] [ ] + z [ ] + l z l 0 a w + l l 0 a w w t [ ] [ ] + t l + l = a t l wt l + l=0 w a + + w 2 t + t l + l = a l t + wt l 2 w + l=0 w t + l t l = w+ 2 +l tw a l t w l=0 Theefoe we wite + 2 +t a ; w + ; ; = a t w+ 2 tw w ; w w ; w a w ; + 4

5 t + t + a t + w+ 2 + tw a w Multiplyig both sides of the above euatio with ;, we have the claimed esult As a coseuece of Theoem, we have the followig Coollay Coollay Fo ay positive itege w, ay ozeo eal umbe a, oegative itege, iteges t ad such that ad t < +, [ ] + 2 +t = a t w+ 2 tw ; a ; w + w ; w w ; w a w ; + 3 Applicatios Now we will give some iteestig coollaies of ou mai esult to Fiboomial sums We stat with a Fiboomial-Lucas coollay : Coollay 2 4+ { 4 + V V + /2 = 2 U 4+! V 4+! 2+ U 2 2+, whee is defied as befoe ad the geealized Fiboacci ad Lucas factoials ae U! = U U U ad V! = V V V, espectively Poof To pove the idetity, if we covet it ito -otatio, the we must pove that 4+ [ ] = ; ; 4+2 If we tae 4 +, a = ad = w = i Theoem, the we wite 4 [ ] t ; + = t + 2 t ; 4+ ; ; ; 4+2 t t + t ; + t If we choose t = 2 whee is oegative itege, the the coditio t < + is satisfied fo = ad so we wite the last euatio as 4 [ ] ; + = ; ; ; ; 4+2 5

6 which euals as claimed 2+ = ; 4+ ; ; 4+2 ; ; 4+2 = ; 4+ ; ; = ; , 2 ; ; 4+2 Coollay 3 Fo > 0, { whee α is defied as befoe U U + U α + = α F +2, Poof Fist we covet the claim ito -otatio Coside [ ] α 2 + α + α α = α α + +2 o o o [ ] [ ] [ ] α = α α + +2 α = = +2, which follows fom Coollay by taig t = = w = ad a = By usig Coollay, we obtai the followig esult: Coollay 4 Let ad l be ay oegative iteges ad c {0,, 2, 3 4 c { 4 c U 4l++ /2 = U 4 c+! V V + 2 V 4 c+! 2 2l U 4 c U2 4l+c, U whee is defied as befoe, ad l fo c {0, 3, ad > l fo c {, 2 Poof We stat by covetig the claim ito -otatio Thus we must pove that 4 c [ ] 4 c 4l+c l 4+ 6

7 which by the biomial theoem euals 4 c [ ] 4 c 2 + 2l = 4 c + 4l+c ; 4 c+ 2 2+l, ; 4 c+ 4l+c+ which, by chagig the summatio ode, euals 4l+c+ 4 c 4l + c + [ ] 4 c s s 4l + c + s s s = 2 2+l 4 c + 4l+c 2 + s 2l = 2 2+l 4 c + 4m+c Now we coside the sums i LHS of the claim ust above: 4 c [ ] 4 c 2 + s 2l , ; 4 c+ ; 4 c+, ; 4 c+ ; 4 c+ whee 0 s 4l + c + Fo =, w = 2, a =, t = s 2l 2, the hypothesis of Theoem is satisfied, that is, t < +, ad so we wite by Coollay, o 4 c [ ] 4 c + 2 s 2l 2 ; 2 2 = s 2l 2 ; 4 c 4 c [ ] 4 c + 2 s 2l = s s 2l 2 ; 4 c 2 ; 2 ; 4 c+ = s ; 4 c 2 ; 2 2 ; 2 2 ; 2 + s 2l 22 2 ; 4 c+ 2 s 2l 2 2 ; 2 ; 4 c+ s 2l 2 ; 4 c+ 2 s 2l 2 ; 4 c+ Fom the last euatio, we wite 4l+c+ 4 c 4l + c + [ ] 4 c s + 2 s 2l 2 s = ; 4 c 2 ; 2 7

8 as claimed 2l 2 ; 4 c+ = ; 4 c 2 ; 2 = ; 4 c + 4l+c+ 2 ; 2 = ; 4 c 2l + 4l+c+ 2 ; 2 = ; 4 c 2l + 4l+c+ 2 ; 2 4l+c+ 4l + c + s 2 2l 2 4l+c+ 4l + c + s s ; 4 c+ s 2l 2 + 4l+c+ 2 2l 2 4l+c+ + 4l+c+ ; 4 c+ ; 4 c+ 2l 2 2 2l c+ ; 4 c+ ; 4 c+ 2 2 c+ ; 4 c+ ; 4 c c + 4 c c+ 2 + ; 4 c+ = ; 4 c 2l + 4l+c+ 2 ; c+ 4 c 2 + ; 4 c+ = 2 2 2l 4 c + 4l+c Coollay 5 Fo > 0, 4+ { 4 + U U + U +3 /2 = ; 4 c+ ; 4 c+, V U 4+2 U 4+2 U 4+3 U 4+4 Poof Fist we covet the claimed idetity ito to -otatio Thus we must pove that 4+ [ ] = ; ; ; 4+2 I ode to cofim this idetity, it is eough to tae 4 +, =, w = 2, a = ad t = 2 i Coollay Refeeces [] G E Adews, R Asey, R Roy, Special fuctios, Cambidge Uivesity Pess 2000 [2] E Kılıç, H Podige, I Aus, H Ohtsua, Fomulas fo Fiboomial Sums with geealized Fiboacci ad Lucas coefficiets, The Fiboacci Quately, 49:4 20, [3] E Kılıç, H Ohtsua, I Aus, Some geealized Fiboomial sums elated with the Gaussia -biomial sums, Bull Math Soc Sci Math Roumaie, 55:03 No 202, 5 6 TOBB Uivesity of Ecoomics ad Techology Mathematics Depatmet Aaa Tuey addess: eilic@etuedut Depatmet of Mathematics, Uivesity of Stellebosch 7602 Stellebosch South Afica addess: hpodig@suacza 8