Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad { } c of the sequeces a ad { d } correspodigly we copute the bioial trasfor of the sequece { ac} i ters of { b } ad { d } I particular we copute the bioial trasfor of the sequeces ( )( ) a ad { } ax i ters of b Further applicatios iclude ew bioial idetities with the bioial trasfors of the products HB H F H L ( x ) ad BF where H B F ad L ( x ) are correspodigly the haroic ubers the Beroulli ubers the Fiboacci ubers ad the Laguerre polyoials Matheatics Subect Classificatio 200: Priary B65; 05A0; 33C45; 40A99 Keywords: Bioial trasfor bioial idetities bacward differece operator haroic ubers Stirlig ubers Beroulli ubers Fiboacci ubers Laguerre polyoials Itroductio ad ai results Give a sequece a its bioial trasfor is the sequece b defied by the forula b with iversio a a 0 () ( ) b 0 The bioial trasfor is related to Euler s series trasforatio [2] ad provides uerous ice ad elegat bioial idetities (see [] [4] [7]) It is a powerful istruet i the theory of special ubers [6] ad i cobiatorics
Our purpose is to develop a techique that helps to geerate ew bioial trasfor idetities fro old Whe the bioial trasfor () is ow we wat to copute the iage sequece 0 ac ( 0 ) where the trasfor d c c d ( ) 0 0 (2) is also ow Wor i this directio was started i the recet paper [] where it was show that a b!!( )( )( ) for ay 0 Writig for brevity have also the followig result fro [] b b b b b for the bacward differece we 0 p p a ( ) b (3) 0 for ay 0 p This will be eeded later I this paper we prove the followig theore Theore Let { a } ad { c } be two sequeces ad let { b } ad { d } be defied by () ad (2) The we have the idetity ac d b 0 0 (4) where b b b with 0 b b For the syetric versio of the trasfor { c } { d } aely d ( ) c c ( ) d 0 0 equatio (4) taes the for 2
a c ( ) d b (5) 0 0 As we shall see with appropriate choices of the sequeces { a } ad { c } this forula produces iterestig ew idetities ivolvig bioial polyoials ad special ubers I several cases the iterated differeces b ca be coputed explicitly The proof of the theore is based o the lea: Lea Suppose the sequeces { a } ad { b } are defied fro () ad let b b b 0 b b The for every two itegers 0 we have ( )( ) a! b (6) 0 or i a shorter for a b (7) 0 The case is (3) with p It was proved i [] The ext lea presets oe possible way to copute the RHS i the above idetity Lea 2 For ay itegers 0 b ( ) b 0 (8) This ca also be put i the for b ( ) b! ( )!( )! The proofs are give i sectio 5 Exaple Let a for all The b 2 0 ad Lea gives for ay 02 ad ay 0 3
( )( )! 2! 2 (9) 0 2 as by a siple coputatio we fid 2 I the ext sectio we apply our theore for the case whe other thigs we cosider the case shall copute the iterated differeces a c ( ) H where H ad prove the idetity x ad i sectio 3 aog H are the haroic ubers We ( ) ( ) H c ( ) H d 0 0 (0) for ay sequece { c } where c ad d are related by (2) I exaples 3 4 ad 5 we apply this forula to the cases where c are correspodigly the Fiboacci ubers the Beroulli ubers ad the Lagurre polyoials A siilar idetity is proved also for the Fiboacci ubers ( ) F c df 2 () 0 0 d with c ad d as above 2 Bioial polyoials For a give sequece a we cosider the polyoials p( x) a x 0 Whe the bioial trasfor () is ow we wat to copute the polyoials p ( x ) explicitly i ters of the ubers p () b We preset here two solutios to this proble They both follow fro Theore with differet choices of the sequeces { a } ad { c } Corollary Assuig the bioial trasfor () is give we have the idetity a x b x x b x x ( x) ( ) 0 0 0 (2) 4
Proof I Theore we set a ( ) x ( ) ( x) 0 so that i view of () b ( x) Siple coputatio shows that for ay z ad ad thus z z ( z ) ( ) ( ) b x x ( x) Fro Theore ( ) x c d( x) ( x) ( ) d x ( x) 0 0 0 Here we chage otatios i order to write this equatio i ters of { a } ad { b } Settig a ( ) c we have fro (2) ( ) d ( ) c a b 0 0 ad the above equatio becoes as eeded a x b x ( x) 0 0 Secod proof idepedet of Theore Usig the iversio forula we ca write p( x) a x x ( ) b 0 0 0 ( ) b ( ) 0 x ad the rest follows fro the well-ow idetity [4] (38) 5
( ) x ( ) x ( x) (22) Rear Idetity (22) itself follows fro Corollary applied to the covolutio idetity ( ) ( ) Here are soe exaples of represetatios of the for (2) Exaple 2 The geeralized Stirlig ubers S( ) of the secod id are defied by the bioial forula (see [3] ad the refereces therei) ( ) ( )! S( ) (23) 0 where is ay coplex uber with Re 0 Accordig to Corollary we have 0 x! S( ) ( ) x ( x) 0 ( ) or chagig x to 0 x x! S( ) x ( x) 0 (24) Whe is a o-egative iteger S( ) are the usual Stirlig ubers of the secod id [6] Exaple 3 Settig x 2 i (2) we obtai the curious idetity 2 ( ) 2 a b 0 0 Exaple 4 With x i (2) 2 2 a 0 0 b which explais the actio of the iterated bioial trasfor 6
The represetatio (2) i the above corollary is short ad siple but its RHS is ot a polyoial i stadard for Fro Theore we obtai also a secod corollary: Corollary 2 Suppose the sequece b is the bioial trasfor of the sequece a The a x b ( x ) (25) 0 0 Proof Taig c x i Theore equatio (4) we have the oe-lie proof d ( ) x ( ) ( x) ( x ) 0 0 Forula (25) gives i fact the Taylor expasio of the polyoial p ( x ) cetered at x Most existig exaples of bioial trasfors with " x " have this forat Exaple 5 With x 2 i (25) we fid 2 a b 0 0 This together with Exaple 3 provide b ( ) 2 b (26) 0 0 Exaple 6 Here is oe very siple deostratio how the corollary wors Let The we have for 0 a ( ) 0 0 ( ) 0 0 ad fro (8) we fid 0 for ad b b ( ) Therefore fro (25) ( ) x ( ) ( x ) ( x) (27) 0 Of course this idetity follows iediately fro the bioial forula 7
3 Idetities with special ubers This sectio cotais soe ew idetities for products of haroic Beroulli ad Fiboacci ubers We start with the followig lea: Lea 3 For ay two itegers ( ) ( ) (3) The proof is give i Sectio 5 Exaple 7 Let fro Lea a ( ) The for ay we have fro the above lea ad ( ) a b (32) The fro Theore the syetric versio equatio (5) ( ) c d (33) This property of the bioial trasfor was discussed i [] It is true for ay two sequeces { c }{ d } related by d ( ) c c ( ) d 0 0 ad the factor ( ) ca be replaced by ( ) For the trasfor () the property is a b Exaple 8 This exaple is related to the previous oe Let H 2 H 0 0 0 be the haroic ubers The we have (see [4]) 8
( ) H (34) Accordig to Lea ( ) H (35) ad the fro Lea 3 for ( ) H (36) ad whe 0 we have 0 H 0 H Therefore fro (34) ad Corollary 2 by separatig the first ter i the secod su ( ) x H( x ) H 0 ( ) ( x ) that is ( ) x ( x) H (37) This equatio also follows fro (27) ad (33) Forula (25) ca be used to evaluate the iterated differeces ow Exaple 9 By iversio i (34) we have 0 b whe the LHS i (25) is ( ) H (38) The versio with " x " was preseted i [2] that is 0 ( ) xh 2 2 x ( x) ( x) ( x) ( x) 2 2 H Coparig this to (25) we coclude that 9
( ) whe 0 ad ( ) H (39) Fro Exaple 9 ad Theore we derive the followig ost iterestig result: Corollary 3 Let { c } ad { d } be ay two sequeces related as i (2) The ( ) ( ) H c ( ) H d 0 0 (30) d For the proof we tae rest follows fro (39) ad Theore a ( ) H ad the i view of () ad (38) we have To show Corollary 3 i actio we shall give several exaples Exaple 0 Applyig property (33) to equatio (38) we fid With the otatio H H ( ) 2 2 2 H 0 2 (2) (2) 2 2 0 we obtai by iversio ( ) b The H (2) ( ) H 0 ad (30) yields (with c H d ( ) H related as i (2)) (2) H H 2 (2) (2) ( ) H H 0 0 (3) Exaple I the sae way startig fro the idetity (see [] equatio (6)) H H ( ) 0 0
ad taig c H ( ) H d i (30) we obtai 2 2 H H H ( ) ( )( ) (32) 0 0 Now we show idetities ivolvig other special ubers Exaple 2 By iversio i (23)! S( ) 0 ad therefore fro (30) with c ad d! S( ) S ( ) H ( )! H S( ) 0 0 ( )! ( ) (33) Exaple 3 For the Fiboacci ubers F it is ow that the followig two bioial idetities are true F F 2 ( ) F (34) 0 F 0 Fro here ad (30) we derive two ew idetities ivolvig products of haroic ad Fiboacci ubers F ( ) H F H F 0 0 (35) by taig c F d ( ) F Also ( ) ( ) HF2 ( ) H F 0 0 (36) with c F2 d F F
Exaple 4 Here we use the Beroulli ubers t t B t 2 t e! 0 For the Beroulli ubers it is ow that B defied by the geeratig fuctio ( ) B B (37) 0 Fro (30) with c ( ) B d B we obtai the idetity ( ) B H B ( ) HB (38) 0 0 Exaple 5 I this last exaple related to (30) we use the Laguerre polyoials x e d x L( x) ( x e )! dx which satisfy the idetity ( x) L( x) 0! Here (30) provides the curious forula ( c L ( x) ad d ( x) )! x x ( ) H L( x) H 0 0 (39)!!( ) Next we tur agai to the sequece of Fiboacci ubers defied by the recurrece F F F 2 ad startig with F0 0 F We ca exted the sequece F for egative idices by usig the equatio F 2 F F Thus we coe to the egatively idexed Fiboacci ubers where 2 F F Coputig the bacward differeces we fid F F F F ( ) 0 2 F F 2 F 3 F 4 2
etc Obviously F F 2 ad this is true for ay o-egative iteger Now we ca forulate the desired result: Corollary 4 For ay pair of sequeces { c } ad { d } as i (2) ad every o-egative iteger we have ( ) F c df 2 (320) 0 0 For the proof we use (4) i Theore with a ad b F (see (34)) ( ) F Forula (320) ca be used i the sae way as (30) to geerate various ew idetifies by choosig differet sequeces { c } For illustratio we provide the followig exaple: Exaple 6 Choosig c ( ) B ad d B where B are the Beroulli ubers we obtai fro (37) ad (320) a idetity coectig Beroulli ad Fiboacci ubers or BF BF 2 (32) 0 0 B F ( F 2 ) 0 0 The Lucas ubers L satisfy the sae recurrece L L L 2 as the Fiboacci ubers ad for the bioial idetities lie (34) hold too Therefore a property siilar to (320) is also true for the Lucas ubers 4 Soe variatios Rear 2 If the bioial trasfor is defied by the forula ( ) a b 0 the (25) taes the for a ( x) b ( x ) (4) 0 0 3
or a x ( ) b ( x ) 0 0 Rear 3 Aother expressio for the coefficiets C( ) b ca be writte i ters of the Stirlig ubers of the first id s ( ) A good referece for these ubers is the boo [6] Suppose the sequece b is the bioial trasfor of the sequece a as i () The C( ) s( ) ( ) b (42)! 0 Proof We have the represetatio! 0 s( ) (see [6]) ad fro here ad (7) C( ) a s( ) a 0! 0 0 s( ) a! 0 0 Therefore i view of (3) we coe to (42) 5 Proofs Here we prove the three leas ad Theore Proof of Lea Whe 0 this is obviously true Tae ay itegers We shall do iductio o Suppose the idetity is true for soe We shall prove it for The LHS the becoes (usig (3) with p i the secod equality) 4
( )( )( ) a ( )( ) a 0 0 ( )( ) a! b! b 0! b b b! ( ) b b!!! b b ( )!! ( )!!!! b ( )! b ( )!! Proof of Lea 2 Fro Lea ad the iversio forula for the bioial trasfor b ( )( ) a a! 0 0 ( ) ( ) b b ( ) 0 0 0 0 0 ( ) by usig the idetity ([7] p5) ( ) ( ) 0 Proof of Theore a c a d 0 0 0 b 5
d a d b 0 0 0 accordig to Lea Proof of Lea 3 The startig poit of this proof is the idetity (22) with We divide both sides by x ad itegrate fro 0 to This yields ( ) ( ) x ( x) 0 dx ( ) ( ) B ( ) The evaluatio of the itegral is fro table [5] aely this is etry 39 (3) Here B( x y ) is Euler s Beta fuctio ( x) ( y) B( x y) ( x y) Refereces [] Khristo N Boyadzhiev Bioial trasfor ad the bacward differece Advaces ad Applicatios i Discrete Matheatics 3 () (204) 43-63 [2] Khristo N Boyadzhiev Haroic Nuber Idetities Via Euler's Trasfor J Iteger Sequeces 2(6) (2009) Article 096 (electroic) [3] Khristo N Boyadzhiev Power su idetities with geeralized Stirlig ubers Fiboacci Quarterly 46/47 (2009) 326-33 [4] Hery W Gould Cobiatorial Idetities Published by the author Revised editio 972 [5] Izrail S Gradshtey ad Iosif M Ryzhi Table of Itegrals Series ad Products Acadeic Press 980 [6] Roald L Graha Doald E Kuth Ore Patashi Cocrete Matheatics Addiso-Wesley Publ Co New Yor 994 [7] Joh Riorda Cobiatorial idetities Robert E Krieger Publ 979 6