Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa Suda Wag School of Mathematical Scieces Ie Mogolia Uivesity ohhot Chia Email: wuyugw@6.com Received 5 Decembe 4; accepted May 5; published 5 May 5 Copyight 5 by authos ad Scietific Reseach Publishig Ic. This wo is licesed ude the Ceative Commos Attibutio Iteatioal Licese (CC BY. http://ceativecommos.og/liceses/by/4./ Abstact Let the umbes P( be defied by ( ( = ( (!! (! ( ( ( ( ( P : = P whee P x x Y x x x ad Y ae the expoetial complete Bell polyomials. I this pape by meas of the methods of Rioda aays we establish geeal idetities ivolvig the umbes P( biomial coefficiets ad ivese of biomial coefficiets. Fom these idetities we deduce some idetities ivolvig biomial coefficiets amoic umbes ad the Eule sum idetities. Futhemoe we obtai the asymptotic values of some summa- P by Daboux s method. tios associated with the umbes ( Keywods amoic Numbes Eule Sum Rioda Aays Asymptotic Values. Itoductio ad Pelimiaies Let Y be the expoetial complete Bell polyomials ad I [] Zave established the followig seies expasio: ( = ( (!! (! P x x Y x x x ( l ( t ( t ( ( ( ( ( = P t = ( ow to cite this pape: W.Y.G.W. ad Wag S.D. (5 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets. Ameica Joual of Computatioal Mathematics 5 96-5. http://dx.doi.og/.46/acm.5.58
W. Y. G. W. S. D. Wag whee ( = fo = = ( = ad Spiess [] itoduced the umbes ( ( =. ( ( ( ( ( P = P P( = fo < ; the Equatio (. is equivalet to whee P( = ( = = P P ( t P ( t = = ( l ( t ( t ( l ( t ( t = ( ( ( ( ( P = ad ( ( ( ( P = ( ( ( ( ( ( P = P ad biomial coefficiets by P ad ivese of biomial coefficiets. Fially i Sectio 4 we give the asymptotic expasios of some summatios dt ht The pape is ogaized as follows. I Sectio we obtai some fo ( meas of the Rioda aays. I Sectio we establish some idetities ivolvig the umbes ( ivolvig the umbes P( by Daboux s method. Due to [] [4] a Rioda aay is a pai ( ( ( of fomal powe seies with h h( ule =. It defies a ifiite lowe tiagula aay ( ece we wite { d } ( dt ( ht ( fuctio of the sequece { } N = ( ( ( d t dt ht d N =. If ( dt ( ht ( is a Rioda aay ad ( f i.e. ( f t = = ft. The we have accodig to the f t is the geeatig d f = t dt ( f( ht ( = t dt ( f( y y = ht ( ( = Based o the geeatig fuctio ( we obtai the ext Rioda aays to which we pay paticula attetio i the peset pape: ( l ( t ( t P( = t t Lemma (see [5] Let α be a eal umbe ad L( z ( α = l. Whe z α α z ( z L ( z l Γ ( α { } m m ( ( (! l ( m z z L z m m. Idetities Ivolvig the Numbes P( ad Biomial Coefficiets Theoem. Let the ( P = P ( = (4 ( 97
W. Y. G. W. S. D. Wag Poof. By ( we have = P ( t = ( l ( t ( t ( l ( ( t t t = ( l ( t = P ( t. = = Compaig the coefficiets of t o both sides of (5 we completes the poof of Theoem. Recall that P( = Thus settig = i Theoem gives the ext thee idetities espectively. Coollay. Let the followig elatios hold = = = = ( = ( ( ( ( ( ( ( ( ( (( ( ( ( =. Theoem. Let the = P ( ( ( = = = Poof. To obtai the esult mae use of the Theoem. Theoem. Let m the = ( m m P ( = P ( mm (7 m Poof. Applyig the summatio popety ( to the Rioda aays ( we have = ( l ( t ( ( l ( ( t m m t P( = t ( y y = t t which is ust the desied esult. Settig m= i Theoem gives the ext Coollay. Coollay Let the = Coollay Let m the = = = t m = t = P( mm m P P m m = m m m = m m m m m ( = ( ( ( ( ( ( ( = ( m m (5 (6 98
W. Y. G. W. S. D. Wag = ( ( ( ( ( ( ( ( m m m( ( ( ( ( ( ( = ( m ( m m m m m m Poof. Settig = i Theoem gives Coollay. Coollay 4. Let the = = = =. = = ( ( ( ( ( = ( ( ( ( ( ( ( = ( (. ( Poof. Settig = i Coollay yields Coollay 4. Theoem 4. Let m the = ( ( ( ( ( ( ( m m P ( = m P ( m m (8 Poof. which is ust the desied esult. Settig m= i Theoem 4 gives the ext Coollay. Coollay 5. Let the = P P ( = ( Coollay 6. The substitutios = i Theoem 4 gives the ext fou idetities espectively. m m = m = m m = m ( m m = m m m = m ( ( ( = m ( ( ( = m ( ( m m ( m m ( (. ( ( ( ( ( ( = ( m m m m ( ( ( ( ( ( m m m m m Settig = i Coollay 5 gives the ext fou idetities espectively. Coollay 7. Let the 99
W. Y. G. W. S. D. Wag = = = = = = ( ( ( ( ( ( ( = ( ( ( ( ( ( ( = ( (. ( Theoem 5. Let m the whee ( = ( ( ( ( ( ( ( ( ( ( ( m P! s = P ( m (9 m! s h ae the Stilig umbes of the fist id. Poof. By ( ad ( we have = ( ( ( t ( ( ( ( ( t ( m P l ( y t l = t y t ( l ( t = = t y( m t t! s = P ( m m! which is ust the desied esult. Settig m= i Theoem 5 gives the ext Coollay. Coollay 8. Let the = ( ( ( P! s = P(! Settig = i Theoem 6 gives the ext Coollay. Coollay 9. Let m the = = = = m = m m m = ( m m ( ( ( ( ( ( ( ( ( = ( m m m m m ( ( m = m (( m ( ( ( ( ( ( ( ( ( m m m m m m ( ( ( m ( (.
W. Y. G. W. S. D. Wag We give fou applicatios of Coollay 9: Coollay. Let the = = = = = = ( ( ( ( ( ( = ( ( ( ( ( 6 = ( ( ( ( ( ( 6 ( (. (. Idetities Ivolvig P( ad Ivese of Biomial Coefficiets Fo idetities ivolvig amoic umbes ad ivese of biomial coefficiets i give i [6]. = I Sectio we obtai some fo P( ad biomial coefficiets by meas of the Rioda aays. Fom these idetities we deduce some idetities ivolvig biomial coefficiets amoic umbes ad idetities elated to ζ ( ζ ( I [7] the ivese of a biomial coefficiet is elated to a itegal as follows = ( t ( t dt ( Fom the geeatig fuctio of P( ad ( we have Theoem 6. Fo be ay itege the P! = = = Poof. Fom ( ad ( we obtai ( ( ( P( = ( ( P t t dt = = = ( ( t l ( ( t t ( ( t ( t ( lt dt! (. = = = This gives (. Coollay Settig = i Theoem 6 The followig elatio holds: = = ( dt
W. Y. G. W. S. D. Wag = ( = ( ( ( ( ( = ( = (4 ( ( = = ( ( ( ( (( ( ( ( Settig = i Coollay gives the ext idetities. Coollay The followig elatio holds ( ( = (5 = (6 = (7 = ( ( ( = (8 = ( ( ( ( = 6 (9 = = ( ( = ( 4 = ( ( ( 5 = ( 8 = ( ( ( ( ( = ( 8 = Coollay. The followig elatio holds ( ( ( ( ( ( 87 = ( 6 ( ( ( = (4 4 ( ( ( = = (5 8 = ( ( ( ( = (6 = 8 ( ( ( ( ( = (7 = 6 ( ( ( Poof. (6 mius( give (4; (7 mius ( (8 mius ( ad (9 mius ( yields (5 (6 ad (7 espectively.
W. Y. G. W. S. D. Wag Leohad Eule (77-78 had aleady stated the equatio Recall the Eule sum idetities [8] [9]. ζ ( s = = p s = p pime 5 = ζ = ζ 4 = ζ = ζ ( ( ( = = 4 = ( = ( The ext we gives idetities elated to ζ ( ζ ( Fo completeess we supply poofs: = = ζ. = = ( ( = = = ( ( s = ( ( = = ( = ( ( ( 4 = = = ( ( ( = ( 4 = = ( = ( ( 7 = = ( 4 (. = ( = ζ = ( ( 6 Similaly we obtai summatio fomulas elated ζ ( ζ ( ( (8 (9 = = ( ζ ( ( = ( ( = ζ ( 4ζ ( ( ( ( By (8 ad (8 (9 ad ( we have ( = ζ ( ( = ( ( ( 4 = ( ζ ( ζ ( ( = ( ( Similaly fo completeess we supply a poof: = = ( ( ( = ( ( = ( ( = = ( ( = = ( ζ ( 4ζ ( = ζ ( 5ζ (. 4 8 = ( ( = ( ( (4
W. Y. G. W. S. D. Wag By (8 mius ( we get = = ( ζ ( (5 = ( ( ( = ( ( = ( ( Applyig (5 ad (4 (6 ad ( we have ( ( 5 = ζ = ( ζ ( ζ ( 8 96 ( = ( ( ( = ( ( ( 4. Asymptotics Theoem 7 Fo be ay itege as we have Poof. By Lemma we have Γ m m ( m P ( m = ( ( m!l ( ( t ( l m / ; m m ( m m l Γ P ( = t m = t ( (!l m ad this complete the poof. Similaly we ca obtai the ext Theoem. Theoem 8. Let be ay itege as we have m P ( m Γ m ( m = m!l m ( ( m Theoem 9. Fo be ay itege as we have { } { } m. l m / ; { } { } m. l m / ; = m!l l m { } { } m. m l l m { }; ( m ( m / m P Γ ( ( m { }. Poof. By Lemma we have = ( ( l ( t t ( m P = ( l ( t m t m l l m { }; m ( m / Γ ( (!l l m m { }. (6 (7 (8 4
W. Y. G. W. S. D. Wag this give (8. Theoem. Fo be ay itege as we have ( ( ( P!l Γ ( l O (9 = = Poof. By Coollay of [] immediately complete the poof of Theoem. Acowledgemets The autho would lie to tha a aoymous efeee whose helpful suggestios ad commets have led to much impovemet of the pape. The eseach is suppoted by the Natual Sciece Foudatio of Chia ude Gat 465 ad Natual Sciece Foudatio of Ie Mogolia ude Gat MS8. Refeeces [] Zave D.A. (976 A Seies Expasio Ivolvig the amoic Numbes. Ifomatio Pocessig Lettes 5 75-77. http://dx.doi.og/.6/-9(76968-5 [] Spiess J. (99 Some Idetities Ivolvig amoic Numbes. Mathematics Computatio 55 89-86. http://dx.doi.og/.9/s5-578-99-769-6 [] Bietze E..M. (8 A Idetity of Adews ad a New Method fo the Rioda Aay Poof of Combiatoial Idetities. Discete Mathematics 8 446-46. http://dx.doi.og/.6/.disc.7.8.5 [4] Wag W. ad Wag T (8 Geealized Rioda Aays. Discete Mathematics 8 6466-65. http://dx.doi.og/.6/.disc.7..7 [5] Flaolet P. Fusy E. Goudo X. Paaio D. ad Pouyae N. (6 A ybid of Daboux s Method ad Sigulaity Aalysis i Combiatoial Asymptotics. The Electoic Joual of Combiatoics. [6] Sofo A. ( Eule Related Sums. Mathematical Scieces 6. [7] Suy B. (99 Sum of the Recipocals of the Biomial Coefficiets. Euopea Joual of Combiatoics 4 5-5. http://dx.doi.og/.6/euc.99.8 [8] Joatha M. (9 Bowei ad O-Yeat Chag. Duallity i Tails of Multiple-Zeta Values 54-4. [9] David B. ad Bowei J.M. (995 O a Itguig Itegal ad Some Seies Relate to ζ (4. Poceedigs of the Ameica Mathematical Society 9-98. [] Flaolet P. ad Sedgewic R. (995 Melli Tasfoms Asymptotics: Fiite Diffeeces ad Rice s Itegals. Theoetical Compute Sciece 44-4. http://dx.doi.og/.6/4-975(948-m 5