EVALUATION OF SUMS INVOLVING PRODUCTS OF GAUSSIAN q-binomial COEFFICIENTS WITH APPLICATIONS

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1 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS WITH APPLICATIONS EMRAH KILIÇ* AND HELMT PRODINGER** Abstract Sums of products of two Gaussia -biomial coefficiets are ivestigated oe of which icludes two additioal parameters with a parametric ratioal weight fuctio By meas of partial fractio decompositio first the mai theems are proved ad the some collaries of them are derived The these -biomial idetities will be trasfmed ito Fiboomial sums as coseueces 1 Itroductio Throughout this paper we use the followig otatios: the -Pochhammer symbol z; ) 1 z)1 z) 1 z 1 ) ad the Gaussia -biomial coefficiets [ ] z; ) z z; ) z; ) Whe z we use the otatios ) ad [ ] istead of ; ) ad [ ] respectively We coveietly adopt the otatio that [ ] 0 if < 0 > Defie the { } seueces as liear recurreces f by p p p With α β p ± p + 4 ) / they admit the followig expressios i the Biet fms α β α β ad α + β Whe α 1+ 5 euivaletly 1 5 )/1 + 5 ) ) the seuece { } is reduced to the Fiboacci seuece {F } ad the seuece { } is reduced to the Lucas seuece {L } F 1 we will use geeralized Fiboomial coefficiets { } 1 1 ) 1 ) with { 0 } { } Whe p 1 they reduces to the usual Fiboomial coefficiet deoted by { } F me details F about the usual Fiboomial geeralized Fiboomial coefficiets their properties ad iterestig appearaces i various places from umber they to liear algebra see [ ] Our approach will essetially be based o the followig coectio betwee the geeralized Fiboomial ad Gaussia -biomial coefficiets { } α ) [ ] 1 with β/α euivaletly α i/ 010 Mathematics Subject Classificatio Primary 11B39; Secodary 05A30 Key wds ad phrases Gaussia -biomial coefficiets Fiboomial ad Lucaomial Coefficiets Sums idetites Partial fractio decompositio 1

2 EMRAH KILIÇ AND HELMT PRODINGER Furtherme we will use geeralized Lucaomial coefficiets { } 1 1 ) 1 ) with { } 1 where is the th geeralized Lucas umber Whe L the geeralized Lucaomial coefficiets are reduced to the Lucaomial coefficiets deoted by { } L : { } L 1 L L L L 1 L L )L 1 L L ) The li betwee the geeralized Lucaomial ad Gaussia -biomial coefficiets is { } α ) [ ] with α α i/ There are may id of sums icludig Gaussia -biomial coefficiets with certai weight fuctios geeralized Fiboomial coefficiets with geeralized Fiboacci ad Lucas umbers as coefficiets f me details see [ ]) Marues ad Trojovsy [16] provide various sums icludig Fiboomial coefficiets Fiboacci ad Lucas umbers F example f positive itegers m ad they show that ad 4m+ j0 4m+ j0 { } 4m j { } 4m + F j F { } 1) jj+1) 4m + L m+1 j ) jj 1) F +4m j 1 F m+ 4m j0 { } 4m j F F F 4+3 F m+1 1) jj 1) L m j Quite recetly Kılıç ad Prodiger [11] compute three types of sums ivolvig products of Gaussia -biomial coefficiets They are of the followig fms: f ay real umber a + 1) + ) a ) ad [ ] + [ ] + 1 [ ] 1) + ) 1 a [ ] 1) ++1 ) a b The they preset iterestig applicatios to geeralized Fiboomial ad Lucaomial sums They prove their results by the partial fractio decompositio method Recetly the method has bee expled i provig various fuctioal idetities f me details we refer to [1 3 4] I this paper we shall ivestigate sums of products of two Gaussia -biomial coefficiets oe of which icludes two additioal parameters with a parametric ratioal weight fuctio Our results will geeralize the results of [11] itroducig two additioal parameters By meas of partial fractio decompositio we shall first prove the mai theems ad the give two collaries from each of them The these -biomial idetities will be trasfmed ito Fiboomial sums as coseueces

3 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS 3 We maily compute five types of the -biomial sums with certai weight fuctios: F ay positive iteger m ad ay real umbers p a ad b m + 1) +1 ) 1 p 1 a [ m + 1) ) a + 1 ] p [ ] ] [ m + 1 b ad 1 [ ] m [ ] m p p p [ ] [ 1) ) 1) ) 1 a a ] [ ] 1) +1 ) 1 a r +c) Note that whe m ad p i the above sums three of them give us the results of [11] By specifyig the parameters oe could derive may differet Fiboomial-Fiboacci-Lucas coseueces The Mai results The results we will preset throughout the paper will be satisfied f a b p w R oegative iteger ad positive iteger m We start with our first result: Theem 1 F oegative itegers m ad such that m m + 1) +1 ) 1 p 1 a ; ) a p; ) a 1 p m ; ) a; ) +1 Proof Cosider the left-had side of the claim m + 1) +1 ) 1 p a p; ) m+ ; ) 1) +1 ) 1 p; ) m + p; ) ; ) ; ) a ; ) p; ) m+ 1 1) 1 +1) 1 p; ) p; ) m + ; ) ; ) a ; ) ) 1 p m + 1 p m+ 1) 1) 1 +1) 1 p; ) ; ) ; ) a without the costat facts cosider p m ) p m 1) 1) 1 +1) 1 ; ) ; ) a p m ) p m 1) 1) ) 1 ; ) ; ) a

4 4 EMRAH KILIÇ AND HELMT PRODINGER ad Defie the fuctios f z) : z pm ) z p m +1) z p m 1) 1 z)1 z) 1 z ) 1 z a A z) : z pm ) z p m +1) z p m 1) 1 z)1 z) 1 z ) 1 Thus set fz) A z) The the partial fractio expasio reads fz) z a p m ) p m 1) ; ) ; ) z ) 1) ) 1 F a) + a z a We multiply this by z ad let z : p m ) p m 1) 0 1) ) 1 ; ) ; ) + F a) a where F a) z pm ) z p m +1) z p m 1) 1 z)1 z) 1 z ) za a pm ) a p m +1) a p m 1) 1 a)1 a) 1 a ) a 1 a 1 p m ) 1 a 1 p m +1) 1 a 1 p m 1) 1 a)1 a) 1 a ) a a 1 p m ; ) a; ) +1 Thus we write p m ) p m +1) p m 1) 1) ) 1 ; ) ; ) a a a 1 p m ; ) a; ) +1 p; ) m+ ; ) 1) ) 1 p; ) p; ) m + ) ) 1 a ; ) a a 1 p m ; ) p; ) a; ) +1 which after some rearragemets gives us m + 1) +1 p as claimed ) 1 1 a ; ) a p; ) As coseueces of Theem 1 we give the followig results: Collary 11 F oegative itegers m ad such that m m + 1) 1 ) 1 + w r+c 1 1) +1 ) ; ) w m+r 1)+c ; ) ; ) w r 1)+c 1 ; ) +1 a 1 p m ; ) a; ) +1

5 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS 5 Proof It is eough to tae p ad a w 1 r c+1 i Theem 1 The the claimed result follows after some rearragemets Collary 1 F oegative itegers m ad such that m m + 1) 1 ) 1 w r +c 1) +1 ) 1 w r 1)+c [ r + c ] 1 w [ ] m + r + c Proof It is eough to tae p ad a w 1 r c i Theem 1 The the claimed result follows after some rearragemets with the defiitios of the Gaussia -biomial coefficiet ad the -Pochhammer otatio w Theem F oegative itegers m ad such that m m + 1) ) p 1) +1 ) [a + p m ] ; ) p; ) Proof Cosider the left-had side of the claim m + 1) ) p ; ) p; ) ; ) p; ) ; ) p; ) ; ) p; ) p; ) m+ 1 1) ) p; ) m + ; ) ; ) ) 1 p m + 1 p m+ 1) 1) ) ; ) ; ) p m ) p m 1) 1) ) ; ) ; ) p m ) p m 1) 1) ) ; ) ; ) without the costat it taes the fm p m ) p m 1) 1) ) ; ) ; ) Now defie the fuctios ad fz) : z pm ) z p m +1) z p m 1) 1 z)1 z) 1 z ) z A z) : z pm ) z p m +1) z p m 1) 1 z)1 z) 1 z ) Thus we see that fz) A z) ) a + 1 z The the partial fractio expasio reads p m ) p m 1) fz) 1) ) ; ) ; ) + C z

6 6 EMRAH KILIÇ AND HELMT PRODINGER We multiply this by z ad let z : where a 1) +1 ) Thus we write as claimed p m ) p m 1) ; ) ; ) 1) ) a + 1 ) + C C 1) p m )+ ) p m ) p m 1) ; ) ; ) 1) ) a + 1 ) a 1) +1 ) + 1) p m )+ ) 1 p m + ) 1 p m+ 1) ; ) ; ) 1) ) a + 1 ) 1) [ a +1 ) + p m )+ ) ] [ ] m + p [ ] 1) ) 1) +1 ) ; ) p; ) [a + p m ] As coseueces of Theem by taig special values of p ad a we give the followig collaries: Collary 1 F oegative itegers m ad such that m m + 1) 1 +1) 1 + w r+c+) 1) +1 ) 1 + w c+1+m+r)) Collary F oegative itegers m ad such that m m + 1 w +r+c ) 1) ) 1) +1 ) [ 1 1) w m+r+1)+c] ; ) ; ) Theem 3 F positive itegers m ad such that m m + 1 1) ) b 1 p a ; ) a 1 a b) m p/a; ) 1 p; ) 1 a; ) +1 Proof Rewrite the LHS of the claim as m + 1 1) ) b 1 p a p; ) m+ 1 p; ) 1 p; ) m + ; ) ; ) ; ) 1) ) b a

7 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS 7 ; ) p; ) 1 ; ) p; ) 1 p; ) m ) ) b p; ) m + ; ) ; ) a ) 1 p m + 1 p m+ ) 1) ) b ; ) ; ) a Without costat facts cosider p m ) p m ) ) ) p m ) p m ) ) ) Now defie the fuctio 1) ) b a 1) ) b a A z) : z pm ) z p m ) z b 1 z)1 z) 1 z ) z a The partial fractio decompositio of Az) taes the fm: p m ) p m ) Az) 1) +1 ) b 1 F a) ; ) ; ) + a 1 z z a Now we multiply this relatio by z ad let z ad obtai p m ) p m ) ) 0 lim 1) +1 ) b z z ; ) ; ) a 1 z ) + F a) z z a which gives us where 0 p m ) p m ) 1) 1 ) b ; ) ; ) + F a) a p m ) p m ) 1) ) b ; ) ; ) F a) a F a) z b) z pm ) z p m +1) z p m )) 1 z)1 z) 1 z ) a b) a pm ) a p m +1) a p m ) 1 a)1 a) 1 a ) a 1) +1 ) + a 1 a b) m p/a; ) a; ) +1 Thus we get p m ) p m ) 1) ) b ; ) ; ) a a 1 a b) m p/a; ) a; ) +1 1) 1 p m +) 1 p m+ ) 1) ) b ; ) ; ) a a 1 a b) m p/a; ) a; ) +1 za

8 8 EMRAH KILIÇ AND HELMT PRODINGER ; ) p; ) 1 ) 1 p m + 1 p m+ ) 1) 1)+ ) b ; ) ; ) a [ ] m as wated p a 1 a b) m p/a; ) a; ) +1 [ ] 1) 1)+ ) b a ; ) a 1 a b) m p/a; ) 1 p; ) 1 a; ) +1 As coseueces of Theem 3 we give the followig collaries by taig special values of p ad a: Collary 31 F positive itegers m ad such that m m + 1 1) 1 ) ) 1 ) Collary 3 F positive itegers m ad such that m m + 1 1) 1 ) ) ; ) ; ) 1 m +1 ; ) 1 ; ) +1 m +1 ; ) 1 1; ) +1 Fially we give the followig theems without proof which could be similarly doe Theem 4 F oegative itegers m ad such that m m 1) ) 1 p a ; ) apm ; ) p; ) a; ) +1 As coseueces of Theem 4 we give the followig collaries: Collary 41 F oegative itegers m ad such that m m 1) +1 ) 1 1 w +r+c 1 [ ] [ 1 r + m + c r + 1) + c 1 1 w r+1)+c 1 Collary 4 F oegative itegers m ad such that m m 1) +1 ) w +r+c ; ) w r 1)+m+c+1 ; ) ; ) w r+c ; ) +1 Now we give our mai last result Theem 5 F oegative itegers m ad such that m m 1) +1 ) a + ) [ a + p m 1)] ; ) p p; ) As coseueces of Theem 5 we have the followig collaries: w ] 1 w

9 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS 9 Collary 51 F oegative itegers m ad such that m m 1) +1 ) 1 1) w r +c) [ 1 w m +r)+c] ; ) ; ) Collary 5 F oegative itegers m ad such that m m 1) +1 ) 1 w r +c) [ 1 w m +r)+c] 3 Applicatios I this sectio we shall give applicatios of our results as the geeralized Fiboomial sums idetities 1) F positive eve positive m ad all itegers r ad c such that m r + 1) c ad r + c 1 1 m + 1) ) 1 1 m r c r + 1) + c 1 +r+c r+1)+c 1 m + 1) ) 1 1 m r c r + 1) + c 1 +r+c r+1)+c ) F positive eve positive m ad all itegers r ad c such that r + c 1 { } 1 { } m + 1) ) 1 1 r + 1) + c 1 m r c +r+c { } m + { } 1) ) 1 +r+c 1 r+1)+c r+1)+c 1 ) {r + 1) + c 1 } 1 { } m r c 3) F all positive odd positive m ad all itegers r ad c such that m r + 1) c ad r 1) + c 1 1 m + 1) ) r + m + c r + c r +c 1) { } m + { } 1) 1 ) 1 r +c 1) r 1)+c 1 r 1)+c { } r + m + c { r + c 4) F positive odd positive m ad all itegers r ad c such that m r + 1) c ad r 1) + c 1 { } 1 { } m + 1) ) 1 1) 1)/ 1 r + c m + r + c r +c r 1)+c ) m + 1) ) 1 {r } { } 1 1) 1 + m + c r + c r +c 1 } 1

10 10 EMRAH KILIÇ AND HELMT PRODINGER 5) F positive m ad all itegers r ad c such that m r + 1) + c 1 ad r + c 1 1 m 1) ) 1 +r+c 1 m + r + c r + 1) + c 1 r+1)+c 1 m 1) ) 1 +r+c 1 m + r + c r + 1) + c 1 r+1)+c 6) F positive m ad all itegers r ad c such that m r + 1) + c 1 ad r + c 1 m 1) ) 1 1 { } 1 { } r + 1) + c m + r + c +r+c { } m { } r+c 1) ) 1 +r+c 1 ) {m } + r + c 7) F odd positive m ad all itegers r ad c m + 1) ) +r+c 1) 1 m+r+1)+c m + 1) ) +r+c 1) 1 m+r+1)+c ad m + m + 1) ) +r+c 1) 1 1 m+r+1)+c { r + 1) + c 1 1) ) +r+c 1) 1 +1 m+r+1)+c 8) F oegative itegers m ad all itegers r ad c such that m m 1) ) r +c m +r)+c m 1) ) r +c m +r)+c 9) F oegative itegers m ad all itegers r ad c such that m m 1) ) r +c ) { m +r)+c if is eve 1 m +r)+c if is odd ad m 1) ) r +c m +r)+c m 1) ) r +c m +r)+c 1 } 1

11 EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS 11 As a showcase we will show how the above geeralized Fiboomial-Lucaomial-Fiboacci Lucas collaries are obtaied from our mai results F this we will prove the first collary that is 1 m + 1) ) 1 1 m r c r + 1) + c 1 +r+c r+1)+c f positive eve First we covert it ito -fm by usig the relatioship give i the Itroductio Thus after rearragig ad simplificatios the claim taes the fm [ ] m + α m+ ) α )[ ] 1) ) 1 α +r+c 1 1 +r+c ) 1 α r+1)+c 1 1 r+1)+c) α m r c ) [ m r c ] [ r + 1) + c 1 α r+1)+c 1 ) m + 1) ) 1 1 +r+c ) α + [ ] [ α c+r) 1 m r c r + 1) + c 1 ) 1 r+1)+c F the case of eve we have to prove that m r+c ) 1) +1 ) [ ] [ c+r) 1 m r c r + 1) + c 1 ) 1 r+1)+c If we tae a r+c ad p i Theem 1 we write m + 1) +1 ) 1 1 +r+c r+c) r c m ; ) r+c ; ) +1 which after some rearragemet is eual to r+c) 1 c+m r+1) ; ) r+c ; ) +1 r+c) ; ) m r c ; ) r+1)+c ; ) r+c 1 ; ) m r+1) c ] 1 ] 1 r+c) 1 ; ) m r c ; ) r+c 1 ; ) 1 r+1)+c ; ) m r+1) c ; ) ; ) r+1)+c 1 1 c+r) 1 m r c r + 1) + c 1 ) 1 r+1)+c as expected So the claim is true ] 1

12 1 EMRAH KILIÇ AND HELMT PRODINGER Refereces [1] CHEN X CH W: Further 3 F 4 ) series via Gould Hsu iversios Itegral Tras Special Fuctios 3 46) 013) [] CH W: A biomial coefficiet idetity associated with Beuers cojecture o Apéry umbers Electro J Combi 111) 004) #N15 3 pp [3] CH W: Harmoic umber idetities ad Hermite Padé approximatios to the logarithm fuctio J Approx They 1371) 005) 4 56 [4] CH W: Partial fractio decompositios ad trigoometric sum idetities Proc Amer Math Soc 1361) 008) 9 37 [5] GOLD H W: The bracet fuctio ad Fouteé Ward geeralized biomial coefficiets with applicatio to fiboomial coefficiets The Fiboacci Quarterly ) 3 40 [6] HOGGATT JR E: Fiboacci umbers ad geeralized biomial coefficiets The Fiboacci Quarterly ) [7] HORADAM A F: Geeratig fuctios f powers of a certai geeralized seuece of umbers Due Math J ) [8] KILIÇ E: The geeralized Fiboomial matrix Europea J Comb 93) 008) [9] KILIÇ E: Evaluatio of sums cotaiig triple aerated geeralized Fiboomial coefficiets Math Slovaca 67) 017) [10] KILIÇ E OHTSKA H AKKŞ I: Some geeralized Fiboomial sums related with the Gaussia - biomial sums Bull Math Soc Sci Math Roumaie 551) 103) 01) [11] KILIÇ E PRODINGER H: Evaluatio of sums ivolvig products of Gaussia -biomial coefficiets with applicatios to Fiboomial sums Turish J Math 413) 017) [1] KILIÇ E PRODINGER H AKKS I OHTSKA H: Fmulas f Fiboomial sums with geeralized Fiboacci ad Lucas coefficiets The Fiboacci Quarterly 494) 011) [13] KILIÇ E PRODINGER H: Evaluatio of sums ivolvig Gaussia -biomial coefficiets with ratioal weight fuctios It J Number They 1) 016) [14] KILIÇ E PRODINGER H: Closed fm evaluatio of sums cotaiig suares of Fiboomial coefficiets Math Slovaca 663) 016) [15] LI N N CH W: -Derivative operat proof f a cojecture of Melham Discrete Appl Math ) [16] MARQES D TROJOSKY P: O some ew sums of Fiboomial coefficiets The Fiboacci Quarterly 50) 01) [17] SEIBERT J TROJOSKY P: O some idetities f the Fiboomial coefficiets Math Slovaca ) 9 19 [18] TROJOSKY P: O some idetities f the Fiboomial coefficiets via geeratig fuctio Discrete Appl Math 15515) 007) * Departmet of Mathematics TOBB iversity of Ecoomics ad Techology Aara Turey address: eilic@etuedutr ** Departmet of Mathematics iversity of Stellebosch 760 Stellebosch South Africa address: hprodig@suacza