PB Sci Bull, Series A, Vol 78, Iss 4, 2016 ISSN 1223-7027 CLOSED FORM EVALATION OF RESTRICTED SMS CONTAINING SQARES OF FIBONOMIAL COEFFICIENTS Emrah Kılıc 1, Helmu Prodiger 2 We give a sysemaic approach o compue cerai sums of suares Fiboomial coefficies wih powers of geeralized Fiboacci ad Lucas umbers as coefficies; he rage of he summaio is o he aural oe bu abou half of i The echiue is o rewrie everyhig i erms of a variable, ad he o use geeraig fucios ad Rohe s ideiy from classical -calculus Keywords: Gaussia -biomial coefficies, Fiboomial coefficies, -aalysis, sums ideiies 11B39, 05A30 1 Iroducio Defie he secod order liear seueces { } ad {V } for 2 by p 1 + 2, 0 0, 1 1, V pv 1 + V 2, V 0 2, V 1 p These recurrece relaios ca also be exeded i he bacward direcio Thus +2 p +1 1 +1, V V +2 pv +1 1 V For 1 ad a ieger m, defie he geeralized Fiboomial coefficie wih idices i a arihmeic progressio by { } m 2m m : ;m m 2m m m 2m m wih { 0 };m { } 1 ad 0 oherwise Whe p m 1, we obai he usual Fiboomial coefficies, deoed by { } ;m Whe m 1, we obai he geeralized Fiboomial F coefficies, deoed by { };1 We will freuely deoe { };1 by { } I his paper, we are ieresed i sums icludig he suare of Fiboomial coefficies of he form { 2 2 +} A addiioal challege is here ha he rage of summaio is o he full rage bu oly abou half of i, amely 0 We maily prese hree ses of ideiies which are expressed i he oio of { } 2 wih m 1, 2 More ;m imporaly, we describe a geeral mehodology how o evaluae hese sums, which will be applicable o may ohers as well Our approach is as follows For a ieger, we use he Bie forms α β α β 1 α 1 1 ad V α + β α 1 + 1 Professor, TOBB iversiy of Ecoomics ad Techology Mahemaics Deparme 06560 Aara, Turey, email: eilic@euedur 2 Professor, Deparme of Mahemaics, iversiy of Sellebosch 7602 Sellebosch Souh Africa, email: : hprodig@suacza 57
58 Emrah Kılıc, Helmu Prodiger wih β/α α 2, so ha α i/ where α, β p ± /2 ad p 2 + 4 Throughou his paper we will use he followig oaios: he -Pochhammer symbol x; 1 x1 x 1 x 1 ad he Gaussia -biomial coefficies [ ] ; ; ; The li bewee he geeralized Fiboomial ad Gaussia -biomial coefficies is { } α m wih α 2 ;m m We recall ha oe versio of he Cauchy biomial heorem is give by +1 [ 2 x ] 1 + x, ad Rohe s formula [1] is [ 1 1 2 x ] x; 1 x All he ideiies we will derive hold for geeral, ad resuls abou geeralized Fiboacci ad Lucas umbers come ou as corollaries for he special choice of Recely, he auhors of [2, 4] compued cerai Fiboomial sums wih geeralized Fiboacci ad Lucas umbers as coefficies For example, if ad m are boh oegaive iegers, he 2 2+1 2 2+1 { 2m 1 P,m 1 m 1 { + 1 2m P,m { V 2m 1 P,m { + 1 V 2m P,m { m 1 4 2, 2 1 m m 1 { m 2+12, 2 } { 2m 1 2 1 m V 4 2, { m V 2+12, 2 as well as heir aleraig aalogues were also preseed, where m V 2 if m, P,m m 1 V 1 2 if < m 1 More recely, Kilic ad Prodiger [3] give a sysemaic approach for compuig he sums of he form: { λ1+r 1 λs+r s i closed form, where r i ad λ i 1 are iegers I his paper we ivesigae sums coaiig he suare of Fiboomial coefficies wih he powers of geeralized Fiboacci ad Lucas umbers, over half he aural summaio rage The approach wors for Fiboacci ad Lucas -ype umbers as facors liewise We discuss boh, he Fiboacci ad Lucas isaces, where he rage of summaios is over all o-egaive iegers i e, abou half of he possible umber of erms For isaces
Closed form evaluaio of resriced sums coaiig suares of Fiboomial coefficies 59 wih he full rage of summaios, we refie ourselves o prese a few represeaive formulæ wihou preseig heir derivaios as well as 2 A sysemaic approach We are ow ieresed o evaluae he followig wo ids of sums { 2 { i 1 m1 V m 2 2 m 3 ad ii 1 m1 m 2 m 3 iii { 2 1 m 1 V m2 m 3 ad iv { 2 1 m 1 m2 m 3 i closed form where m i are iegers The sums of he ype i will be raslaed io -oaio: [ 2 2 2 1 m1++ i m 3m 2 1 2 m 3m 2 1 + m 3 m2 For our mehod o wor, firsly he facor 1 mus appear ad secodly he erm for mus coicide wih he erm for Tha meas ha we have wo possibiliies for he firs codiio: m 2 is eve, m 3 is eve such ha 2m 1 m 2 m 3 The we are able o evaluae he sums { 2 i closed form Their -forms are 1 2 1 rs V 2s r ad { 2 1 rs V2r s [ 2 1 2 rs 1 + r 2s ad [ 2 1 2 1 2 rs 1 + 2r s, 1 respecively Now we will examie wheher he secod codiio is saisfied We cosider he erm for ad igore cosa facors, [ 2 [ 2 1 2 rs 1 + r 2s 1 2 +rs 1 + r [ 2s 2 which is he erm for Thus we have ha [ 2 1 2 rs 1 + r 2s [ 2 2 2s 1 + 1 2 1 2 rs r + 1 2s, [ 2 1 2 rs 1 + r 2s
60 Emrah Kılıc, Helmu Prodiger Now we compue he sum over he full rage: [ 2 of he form 2 1 2 rs 1 + r 2s [ 2 1 2 rs 1 + r 2s 1 2 +rs 2 2s 1 2 +rs [ 2 2s 2s 1 2 2 rs r 2 2s r [ 2 1 2 2 rs+r 2s 1 2 +rs 2s 2 [ 2 r 1 2 2+rs Thus, coceraig o he ier sums, we have o evaluae a fiie umber of erms 2 [ 2 1 2 2+µ, where µ r s is a ieger Now we will explai how his ca be doe Cosider 2 [ 2 1 2 2+µ 22 + [ z 2] 2 22 + [ z 2] 2 2 2 z ] [ 2 2 z 22 + [ z 2] z; 2 z µ ; 2 2 2 Summarizig, he sum of ieres is evaluaed as [ 2 1 2 rs 1 + r 2s 2 2s 1 [ 2 + 1 1 2 2s 2 +rs+1 2 1 2 2 µ z 2 2 1 2 µ z 2s r [ z 2] z; 2 z µ ; 2, where µ is defied as before I order o evaluae [ z 2] z; 2 z µ ; 2, we observe ha here are facors 1 z i ad 1 + z i ha ca be combied o 1 z 2 2i Tha is he reaso ha we eed he facor 1 i our sums, as meioed before I fac, here are 2 µ such pairs, ad oly 2 µ separae facors They mess up he fial resul, bu sice µ is a cosa o depedig o, we sill ge a closed from evaluaio We have o evaluae a fiie umber of erms of he form [z 2 ]z a b z 2 c ; 2 2 µ b [z 2 a ]z 2 c ; 2 2 µ This is eiher 0 for 2 a odd or c2 a b+ 2 µ 2 2 a 1 a 2 a/2 a/2 1 2 2
Closed form evaluaio of resriced sums coaiig suares of Fiboomial coefficies 61 oherwise Eveually we ed up wih a fiie liear combiaio of erms of he form [ µ a/2 for some iegers µ ad a The fial sep is o raslae such a resul bac o expressios i erms of 2 µ ad simplify accordig o he Bie formula relaed o he recursio of a/2 ;2 secod order for For 1, he secod par of i, we ca chec i he same way ha our wo codiios are saisfied Thus we may wrie [ 2 1 2 rs 1 + 2r s [ 2 2 s 1 + 1 [ 2 1 2 rs 1 + 2r s 2 Now, similar o he previous case, we wrie [ 2 1 rs 1 + 2r s + 2 +rs 1 s 2 s 2 2r [ 2 1 2 2+rs 2, which, by usig our previous resul, euals s 2 +rs+1 1 s 2r [ z 2] z; 2 z µ ; 2, where µ r s 2 Therefore we have [ 2 [ 2 where µ r s 2 1 2 rs 1 + 2r s 2 s 1 + 1 s +rs+1 1 2 2 s 2r [ z 2] z; 2 z µ ;, 2 Now we move o sums of ype ii ad raslae hem io -oaio: [ 2 1 m2 2 2 1 m1+ i m 2m 3 1 1 2 m 2m 3 1 1 m m 3 2 For our mehod o wor, we reuire ha m 2 is eve such ha 2m 1 m 2 m 3 The we are able o evaluae he sums { 2 1 rs r 2s i closed form I -oaio, we have o evaluae [ 1 2s 1 s 2 s 2 1 rs 1 r 2s
62 Emrah Kılıc, Helmu Prodiger Sice he erm for 0 evaluaes o 0 ad we have agai symmery bewee ad we ca wrie [ 2 1 rs 1 r 2s 1 [ 2 1 rs 1 r 2s 2 Now we deal wih he full rage summaio [ 2 1 rs 1 r 2s 2s [ 2 2s 1 2s 1 rs which, by aig isead of, euals 2s 1 r [ 2 1 rs 2s 1 2s 2 2 1 rs Thus, we agai ecouer erms of he form [ 2 1 2 2+µ, [ 2 1 2 2+rs where µ r s is a ieger The reame of hese erms is covered by he previous discussio Summarizig, our evaluaio aes he form [ 2 where µ is defied as before 2 rs 1 1 r 2s 1 2s 2 1 2 + 2s 1 µ [ z 2] z; 2 z µ ; 2, I he remaiig secios, his geeral program will be illusraed i more deail o four examples Furher, we will lis several aracive formulæ ha were obaied usig he procedure jus described Fially we prese resuls o he sums wih full summaio rage 3 Illusraive Examples Now we wor ou four examples ha fall io he geeral scheme meioed above i more deail Also we will prese some addiioal examples wihou proof Theorem 31 For > 1, { 2 { 2 V2 2 2 ;1 { + ;1 ;2 { 2 + 2 ;2 + 2 2 + V 2 { 2 2 1 }, ;2
Closed form evaluaio of resriced sums coaiig suares of Fiboomial coefficies 63 where is defied as before Proof Firs we cover he lef-had side of he claim i -oaio: { 2 [ 2 V2 2 α 22 2 α 4 1 + 2 2 [ 2 α 22 [ 2 α 22 α 22 +4 1 + 2 2 1 2 1 + 2 2 Secod we cover he righ-had side of he claim i -oaio, sippig deails: { { } { } 2 2 2 2 2 + 2 + + 2 2 { } 2 2 2 + V 2 ;1 ;2 ;2 1 ;2 [ 2 2 2 2 α 2 22 + 2 2 + 2 2 1 1 2 2 2 1 1 + 2 2 2 1 So we eed o prove ha [ 2 1 2 2 1 + 2 [ 2 2 2 + [ 1 2 2 [ 2 + 2 2 2 1 2 2 2 1 1 + 2 [ 2 2 1 Thus, accordig o our approach, we wrie [ 2 1 2 2 1 + 2 2 [ 2 2 + 1 1 2 +3 2 2 [ z 2] z; 2 2 2 z 21 ; 2 [ 2 2 + 1 1 [ 2 2 +3 z 2 ] z; 2 z 2 ; 2 + 2 2 [ z 2] z; 2 z; 2 + 4 [ z 2] z; 2 z 2 ; 2 [ 2 2 + 1 1 [ +3 z 2 ] z 2 ; 2 2 2 2 2 1 + z 2 1 1 + z 2 2 1 z/ 2 1 z/ + 2 2 [ z 2] z 2 ; 2 + [ 2 4 z 2] 1 + z 1 + z 1 z 2+1 1 z 2 z 2 ; 2 2 z 2 ; 2 2 [ 2 2 + 1 1 2 2 +3 2 2 [ z 2] z 2 ; 2 + [ z 2] z 2 ; 2 2 2 2 1 + 3 1 2 1 2 2 2+1 + 4 z 2 + 4 6 z 4 + 4 [ z 2] ] 2
64 Emrah Kılıc, Helmu Prodiger which, by Rohe s formula ad afer some rearragemes, euals [ 2 2 + 1 1 2 2 +3 2 2 2 1 2 2 2 1 2 2 2 2 2 2 2 + 4 6 1 2 2 2 2 2 3 1 2 1 2 2 2+1 + 4 2 2 1 1 2 2 1 2 ] + 4 1 2 2 +4[ 2 2 2 ] + 4+2 1 2 2 +4 8[ 2 2 2 2 2 1 2 1 2 2 2+1 + 4 1 ] 1 2 2 as claimed [ 2 2 [ 2 + 2 2 + 2 [ 2 1 1 2 2 2 1 1 + 2 [ 2 2 1 ] 2, +4 4[ 2 2 1 2 Theorem 32 For > 1, { 2 1 V 2 ;1 { { } 2 + 2 1 2 1 ;1 ;2 Proof Firs we cover he lef-had side of he claim i -oaio: { 2 [ 1 2 V 2 1 2 1 2 1 + 2 ;1 Secod we cover he righ-had side of he claim i -oaio: { { } 2 + 2 1 2 1 1 2[ 2 2 1 + 2 ;1 ;2 Thus we eed o prove ha [ 2 1 2 1 + 2 [ 2 2 1 + 2 2 Thus, accordig o our approach, we wrie [ 2 1 2 1 + 2 [ 2 + 1 +2 1 2 2 [z 2 ] z; 2 z 1 ; + [ 2 2 z 2] z; 2 z; 2 [ 2 + 1 +2 1 [ z 2 ] z 2 ; 2 2 2 2 1 1 z/ 1 + z 2 1 + 2 [ z 2] z; 2 1 z; 2 1 1 + z 1 z 2 2
Closed form evaluaio of resriced sums coaiig suares of Fiboomial coefficies 65 [ 2 [ 2 + 1 2 1 2 1 2 2 + 2 1 2 [ 1 + 2 as desired; we sipped a few simple iermediae seps for breviy Theorem 33 For oegaive, { 2 + 2, 2 2 1 2 { 1 1 ;2 Proof Followig he program oulied before, we eed o prove ha [ 2 1 2 1 1 1 2 1 2 + 1 2 Thus, accordig o our approach, we evaluae he sum as follows, oly givig some ey seps: [ 2 2 1 1 2 1 2 1 2 + [ z 2] z; 2 z/; 2 2 [ z 2] z; 2 z; 2 + [ z 2] z; 2 z; 2 1 2 1 2 + [ z 2] 1 z/ z; 2 1 z; 2 1 1 + z 2 1 2 [ z 2] z 2 ; 2 2 + [ z 2] 1 + z z; 2 1 z; 2 1 1 z 2 1 ] 2 1 2 + 2 1 2 1 2[ +1 2 1 2 2 2 which, by some simple rearragemes, euals 1 2 2 1 as claimed 2, Theorem 34 For oegaive, { { 2 2 2 2 2 2 2 1 2 + 1 Proof The ideiy i -form is [ 2 + Thus we wrie [ 2 1 2 1 2 2 2 2 1 1 2 2 } ;2 1 1 + 1 2 1 1 2 2 2 1 2
66 Emrah Kılıc, Helmu Prodiger 1 2 1 2 + 2 [ z 2] 1 z/ 2 1 z/ 1 + z 2 1 1 + z 2 2 z; 2 2 z; 2 2 2 [ z 2] z 2 ; 2 2 + [ 2 z 2] 1 + z 1 + z 1 z 2 1 z 2+1 z 2 ; 2 2 z 2 ; 2 2 1 2 1 2 + 2 [ z 2] z 2 ; 2 2 2 1 + 3 1 2+1 2 1 2 2 + 4 z 2 + 4 6 z 4 2 [ z 2] z 2 ; 2 2 + 2 [ z 2] z 2 4 ; 2 2 2 which, by Rohe s formula, euals 2 2 2 2 2 as claimed 2 1 + 1 1 2 1 1 2 [ 2 2 1 1 1 2 1 2 2 2+1 + 4 2 ] 2, [ 2 1 2 Now we will prese a few addiioal resuls wihou explici proofs; hey ca be doe i exacly he same way as he previous examples: Theorem 35 1 For > 1, { 2 2 4 2 { 2 2 + 3 2 4 + 2 2 4 V1 2 2 1 2 3 2 2 + 2 2 2 2 For oegaive, { 2 4 3 +1 + 3 1 + V 2 1 V 2 { ;2 To fiish, we prese wo examples where he sums are over he full summaio rage Theorem 36 1 For oegaive, { 2 + 2 For > 0, V 2 1 4 1 V 2 { ;2 { 2 V 2 1 2V { } 4 2 2 2 + 1 ;2 R E F E R E N C E S [1] G E Adrews, R Asey, R Roy, Special fucios, Cambridge iversiy Press, 2000 [2] E Kılıç, H Prodiger, I Auş, H Ohsua, Formulas for Fiboomial Sums wih geeralized Fiboacci ad Lucas coefficies, The Fiboacci Quarerly 49:4 2011, pp 320 329 [3] E Kılıç ad H Prodiger, Closed form evaluaio of sums coaiig suares of Fiboomial coefficies, acceped by Mahemaica Slovaca [4] E Kılıç, H Ohsua, I Auş, Some geeralized Fiboomial sums relaed wih he Gaussia -biomial sums, Bull Mah Soc Sci Mah Roumaie, 55:103 No 1 2012, pp 51 61 } ;2