Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions. Donal F.

Transcription

1 Fourier erie repreeio of he logrihm of he Euler gmm fucio d he Bre muliple gmm fucio Dol F. Coo dcoo@bopeworld.com 5 Mrch 9 Abrc Kummer Fourier erie for log Γ ( ) i well ow, hvig bee dicovered i 847. I hi pper we develop correpodig Fourier erie for logrihm of he Bre double gmm fucio (d he mehod my be eily eeded o he higher order muliple gmm fucio). Some pplicio of hee Fourier erie re eplored.. Iroducio We recll he He ideiy for he Hurwiz ze fucio [7] which hold for ll C ecep = ( ) (.) ς (,) = + = ( + ) d wih hi become (.) ς (, ) = ( ) ( + ) + = We hve he well-ow Hurwiz formul for he Fourier epio of he Hurwiz ze fucio ς (, ) repored i Tichmrh reie [38, p.37] co i (.3) ς (,) = Γ( )i + co ( ) ( ) where Re ( ) < d <. I, Boudjelh [] howed h hi formul lo pplie i he regio Re ( ) <. I my be oed h whe = hi reduce o Riem fuciol equio for ς ( ). Leig we my wrie (.3) (.4) co i ς (, ) = Γ ( ) co + i ( ) ( )

2 co[ / ] = Γ( ) ( ) which i vlid for (σ <, < ; < σ, < ) The derivio of (.3) h bee implified by Zhg d Willim [4].. Kummer Fourier erie repreeio of he gmm fucio Muliplyig (.4) by, we ee h (.) (.) co[ / ] f(,) = ς (,) = Γ ( + ) ( ) = ( ) ( + ) + = Differeiio of (.) reul i (.3) p p f (,) = ()( + )log( + ) p + = d we hve he priculr vlue = (.4) p p f (,) = ( )( + )log( + ) p = + = = We lo hve from (.) (.5) ( / )i[ / ] co[ / ]log( ) f(, ) = Γ ( + ) Γ ( + ) ( ) ( ) + Γ ( + ) co[ / ] ( ) d hu f(,) () co i log( ) i = +Γ = = = =

3 We oe he fmilir rigoomeric erie how i Crlw boo [3, p.4] (elemery derivio re lo coied i [8]) (.6) (.7) co log(i ) = ( < < ) i = ( < < ) Uig hee ideiie reul i i log( ) f(, ) = log(i ) + [ γ log( )] = d we he hve ( ) ( + )log( + ) + = i log( ) = log(i ) + [ γ log( )] We howed i [4] h (.8) log Γ ( ) = ( ) ( + )log( + ) + + log( ) + = d we herefore obi Kummer Fourier erie [9] for he log gmm fucio (which, becue we relied o (.6) d (.7), i oly vlid for < < ) (.9) log log Γ ( ) = log + [ γ + log( )] + i i Referece o (.6) d (.7) cofirm h (.9) i properly decribed Fourier erie epio for log Γ( ). I 985 Berd [9] gve elemery proof of hi Fourier erie epio, which w origilly derived by Kummer i 847. The erie immediely give rie o he fmilir reul (.) log Γ = log 3

4 I my be oed h i i o poible o differeie (.9) becue, i eily ee, he reulig ifiie erie i diverge. Nowihdig hi, rigoomeric epio (which i o Fourier erie) ei for he digmm fucio ψ ( ) how i (8.7) below. Alerively, differeiig (.) give u log( + ) (.) ( ) ς (, ) + ς(, ) = ( ) ( ) = + = + d evluio = produce (.) ς (, ) = ς(, ) + ( ) ( + )log( + ) + = We hve he well ow reliohip bewee he Hurwiz ze fucio d he Beroulli polyomil B ( ) (for emple, ee Apool boo [5, pp ]) (.3) (, ) B () m+ ς m = m + for m N o d i my be oed h hi ideiy my lo be deduced from (.) becue we hve how i [9] (.4) Bm+ () = ( )( + ) + = m+ I priculr, from (.3) we hve ς (, ) = B ( ) = From (.) we he ee h (.5) ς (, ) = + ( ) ( + )log( + ) + = d, comprig hi wih (.8), we hve herefore deduced Lerch ideiy [9] (.6) ς (, ) = log Γ( ) log( ) 4

5 Sice ς (,) = ς () = log( ) hi my be epreed ς (, ) ς () = log Γ ( ) Lerch eblihed he bove reliohip bewee he gmm fucio d he Hurwiz ze fucio i 894 (oher derivio re coied i, for emple, Berd pper [9] d [4]). A oed by Berd [9] we hve wih i (.9) (.7) log log Γ( ) = log [ γ + log( )] + i i d hu ddig (.9) d (.7) ogeher we ee h log Γ ( ) + log Γ( ) = log i which i imply Euler reflecio formul for he gmm fucio (.8) Γ() Γ( ) = i Differeiig (.8) reul i ψ ( ) = ( ) log( + ) + ( ) + = + = d, ice ( ) = δ, =, we ee h [4] (.9) ψ ( ) = ( ) log( + ) + = Thi reul w lo recely obied i differe wy by Guiller d Sodow [4]. Differeiig (.9) give u (.) ( ) ψ () = + = ( + ) 5

6 d, ee from (.), hi i equl o ς (, ). ( ) ( Sice i / = i 3 / ), leig = /4 d = 3/4 repecively i (.9) d ddig he wo equio ogeher, we obi 3 log Γ + log Γ = log + log 4 4 which of coure my lo be eily obied from Euler reflecio formul (.8) for he gmm fucio (or, lerively, from Legedre duplicio formul for he gmm fucio [36, p.7]). Noig h i ( / ) = i ( 5 / ) uforuely doe o i u becue = 5/4 fll ouide of he regio of vlidiy of Kummer formul (.9). 3. A pplicio of Prevl heorem Applyig Prevl heorem [8, p.338] o he Fourier erie (.9) we hve (3.) log ς () log Γ( ) log [ γ + log( )] d i = = or equivlely we hve he i compoe iegrl log Γ ( ) d+ log d+ [ γ + log( )] d 4 i log Γ ( )log d+ [ γ + log( )] log i i d ς ( ) [ γ + log( )] log Γ ( ) d = I order o deermie ur. log Γ( ) d we ow evlue he l five of hee iegrl i 6

7 Secod iegrl We hve log d = (log log i ) d i = log log log id+ log id The Log-Sie iegrl L ( ) θ re defied for by (3.) θ L ( θ ) = log i d d hee iegrl hve bee coidered by my uhor, icludig Beumer [], Lewi [3], Boro d Moll [, p.45] d Srivv d Choi [36, p.8]. From [3] we hve for emple (3.3) L ( ) = log i d = (3.4) 3 L 3( ) = log i d = Wih he ubiuio = we ee h log i d = log i [ ] d wih he ubiuio = y we hve d hece log[ i] d = log[ i ] d y dy 7

8 [ d ] [ ] log i = log i d= Thi give u he well-ow iegrl (ee lo (5.) below) (3.5) log i d= log Similrly we hve d d d log i = log [ i ] = log [ i ] d we he ee from (3.4) h (3.6) log [ i ] We hve d = [ ] log i d = log id+ log log id+ log d herefore uig (3.6) we hve (3.7) log i d= + log Thi i he problem publihed by Brememp [] i 957. Alerively, we could lo pply Prevl heorem o (.6) d obi log (i ) d = = Applyig Prevl heorem o (.7) reul i d = = 8

9 I i eily ee h d = d we herefore obi Euler formul for ς () = 6 Thi mehod could lo be pplied o he fifh d ih iegrl. To coclude hi pr we hve log d = (log log i ) d i = log log log id+ log id = log + log log + + log = log ( ) + Third iegrl The hird iegrl i rher bic bu, geerliio, we oe [5, p.76] + (!) () = ( ) B d B ( )! d hu d = Fourh iegrl We howed i equio (6.3) of [8] h 9

10 (3.8) [ ] i( ) Si( ) log Γ ( + ) log i( ) d= = ς () 4 where Si() i he ie iegrl fucio defied by [3, p.878] d [, p.3] i Si( ) = d, Si () = We hve he well-ow iegrl from Fourier erie lyi i = d d herefore defiig i( ) = Si( ) we hve i i i i( ) = d d = d I equio (6.7j) of [8] i w lo how h Si( ) (3.9) log A = 4 where A i he Gliher-Kieli co Therefore we hve log A = ς ( ) (3.) [ ] I i eily ee h log Γ ( + )log i( ) d= log A ς() 4 4 [ ] ( )( log Γ ( + )log i( ) d= log + log Γ ( ) log + logi( )) d

11 = log log d+ log log Γ ( d ) + log logi( d ) + log Γ( )logi( d ) = log + log log( ) + log logi( d ) + log Γ( )logi( d ) where we hve ued Rbe iegrl log Γ ( d ) = log( ) which my lo be obied direcly from Aleeiewy heorem (4.8) below. Uig (.6) we hve co log logi( d ) = log log d log d log log co d = where we hve umed h i i vlid o ierchge he order of iegrio d ummio. Le u ow coider he iegrl u i i log.co d = log d u u We hve i lim log i = lim log = Therefore we ge u i u log u i log.co d = d u

12 u i u logu i = d d hece we ge u i u log u Si( u) log.co d= Priculr ce re follow u i ulog u Si( u) log.co d= Si( ) log.co d= Hece we hve (3.) Si( ) log logi( ) d= log + = = log A + log 4 Therefore we obi (3.) log Γ ( )logi( ) d= log log( ) 4 which w previouly deermied by Epio d Moll [] i very differe mer. We could lo hve employed he geerlied Prevl heorem [8, p.343] f( ) g( ) d α ( α bβ) = + + o evlue (3.) by uiliig he ow Fourier erie for he wo compoe of he iegrd. Fifh iegrl We ee h

13 log d = log d B ( ) log i i + d d hi i priculr ce of iegrl oed by Epio d Moll [] (3.3) (3.4) B ()logi d + = ( ) ( )! ς (+ ) B ()logi d = ( ) Very elemery proof of he bove iegrl re give i [8] where we ued he bic ideiy b (3.5) p( )co( α /) d b = p( )iαd which, how i [8], i vlid for wide cl of uibly behved fucio. Specificlly we require h p( ) i wice coiuouly differeible fucio. I hould be oed h i he bove formul we require eiher (i) boh i( / ) d co( / ) hve o zero i [ b, ] or (ii) if eiher i( / ) or co( / ) i equl o zero he p() mu lo be zero. Codiio (i) i equivle o he requireme h i h o zero i [ b, ]. We he hve log i d = I fc hi c be how much more direcly by uig he ubiuio = / he oig h he iegrd of he reulig iegrl i odd fucio. d Thi give u he fifh iegrl log d = i Sih iegrl We oe h 3

14 Γ d = B Γ log ( ) ( ) log ( ) d which i priculr ce of iegrl oed by Epio d Moll [] (3.6) (3.7) B ς ( ) B ( )log Γ ( ) d = log( ) γ ς ( ) + ( ) ( )! ς (+ ) B ()log Γ () d = = ς ( ) ( ) d we herefore obi ς () log Γ ( ) d = log( ) γ ς () Uig (4.5) we hve ς () = log( ) + γ + ς ( ) ς () d hu (3.8) log Γ ( ) d = ς ( ) = log A The bove collecio of he bove five iegrl ow eble u o evlue he fir iegrl. Fir iegrl Uig he bove we deermie h (3.9) () γ γ γ ς () ς log Γ ( ) d = + + log( ) + log ( ) [ + log( )] which Epio d Moll [] lo howed, dmiedly wih much le effor. 4. Fourier erie repreeio of he Bre double gmm fucio We hve from (.5) i he ce where = 4

15 f(, ) log( ) i co = + = = = co log 3 co + γ = = From (.3) we oe h f (, ) = ( ) ( + ) log( + ) = + = The Bre double gmm fucio Γ ( ) / G ( ) = defied, ier li, by [36, p.5] (4.) G ( + ) = ( ) ep ( γ + + ) ep + = d i i eily ee h G () =. I w lo how i [4] h he Bre double gmm fucio could be epreed he logrihmic erie (4.) + log G( + ) = ( ) ( + ) log( + ) + log Γ ( ) + B ( ) + ς ( ) = 4 where B ( ) re he Beroulli polyomil. We herefore hve he rigoomeric erie (4.3) i 3 co log G( + ) = + log( ) + γ 4 co log + + log Γ ( ) + B ( ) ( ) 4 + ς which my be epreed Fourier erie by oig h [4, p.338] (ee lo (7.9) d (7.9b) below) co B () = ice 5

16 (4.4) (4.4b) co B = N, N =,,... N () N + ( ) ( )! N ( ) i B = N +, N =,,,... N + N + () ( ) ( )! N + ( ) To obi pure Fourier erie for log G( + ) i would lo be ecery o deermie he Fourier erie epio for log Γ ( ) uig Kummer ideiy (.9). Wih = i (4.3) we hve ice G() = G() Γ () = () = log( ) + +ς ( ) (4.5) ς ( γ ) which my lo be eily derived by differeiig he fuciol equio for he Riem ze fucio (ee for emple [9]). Sice lim[ log Γ ( )] = lim[ log Γ ( + ) log ] =, i my be oed h equio (4.3) lo pplie whe = d hi lo reul i (4.5). Leig = / i (4.3) give u 3 ( ) ( ) log log G(3/ ) log( ) log ( ) γ = ς The lerig Riem ze fucio i defied by ( ) ς () = + d i i eily ee h ς () = = = ( ) (+ ), (Re () > ) + ( ) = = ς (), (Re ) > ; ) ( We he hve ( ) = ς () = 6

17 Differeiig give u ς () = ( )() ς ς ς ς = + () ( ) () ()log d hu ( ) log ς () = = () + ()log ς ς We he obi 3 log G(3/ ) = log( ) + γ + ς () + log + log + ς ( ) Uig (4.5) hi become (4.6) 3 log G(3/ ) = log + log + ς ( ) 4 4 Sice [36, p.5] G( + ) = G( ) Γ( ) we ee h [36, p.6] (4.7) 3 log G(/ ) = log log + ς ( ) 4 4 origilly deermied by Bre [7] i 899. Uig (.6) d iegrig (.9) reul i i log Γ ( ) d = log + + log [ γ + log( )][ B ( ) B ()] 4 log log co + 7

18 d comprig hi wih (4.3) we ge log Γ ( d ) = log( ) [ γ + log( )][ B( ) B()] 3 co log G( + ) + log( ) γ + = + log Γ ( ) + B ( ) + ς ( ) + log 4 Uig (4.5) d ome lgebr, we obi Aleeiewy heorem [36, p.3], furher derivio of which i coied i equio (4.3.85) of [4] (4.8) log Γ ( d ) = ( ) + log( ) log G( + ) + log Γ( ) I imilr wy, oe could iegre (4.) o obi he iegrl log G( + ) d. 5. The Goper/Vrdi fuciol equio Wih (5.) = i (.) we obi ς = ς = (, ) (, ) ( ) ( ) log( ) Therefore, uig (.7) we ee h ς = (, ) B ( ) ( ) ( ) log( ) 4 = d ubiuig (4.) we obi (5.) log G( + ) log Γ ( ) = ς ( ) ς (, ) Thi fuciol equio w derived by Vrdi i 988 d lo by Goper i 997 (ee Admchi pper [3]). Leig i (5.) give u log G( ) ( ) log Γ( ) = ς ( ) ς (, ) 8

19 Noig h l og G( ) = log G( ) + log Γ( ) we obi (5.3) log G( ) + log Γ( ) = ς ( ) ς (, ) Leig i (4.3) give u i 3 co log G( ) = + log( ) + γ 4 d hece we hve co log + + ( )log Γ( ) + ( ) ( ) 4 B + ς G( + ) i log = + log[ Γ( ) Γ( )] + [ B ( ) B ( )] G( ) 4 Uig he well-ow propery of he Beroulli polyomil [36, p.6] we he hve B ( ) = ( ) B ( ) G( + ) i (5.4) log = + log[ Γ( ) Γ( )] G( ) = Thi my be wrie log G( + ) log Γ( ) [log G( ) + log Γ( )] = i = d uig he Goper/Vrdi ideiie (5.) d (5.4) we ee h i = (5.5) ς (, ) ς (, ) previouly oed by Admchi []. Uig (.6) we my wrie (5.4) = G( + ) log = log(i ) d + log logi( ) G( ) 9

20 which give u (5.6) G( + ) i log = log + log(i ) G( ) d d uig iegrio by pr, we ee h hi i equivle o he followig iegrl formul origilly foud by Kieli [36, p.3] i 86 (5.7) G( + ) log = log( ) co G( ) d (which i recorded eercie i Whier d Wo [39, p.64]). Thi my lo be wrie for < (5.8) co d= ς (, ) ς (, ) + log( i ) which w lo derived i equio (4.3.58) i [4] by differe mehod. Iegrio by pr reul i cod= logi logid d we obi (5.9) log(i ) d = [ ς (, ) ς (, )] Wih = / i (5.9) we redicover Euler iegrl (5.) log i d= log From (4.3) d (4.4) we hve

21 i co log G( + ) log Γ( ) ς ( ) = + ( log( ) + γ ) 4 co log + d (5.) herefore give u he Fourier erie for ς (, ) (5.) i co colog ς (, ) = ( log( ) + γ ) 4 which w derived i differe wy i equio (4.4.9i) of [7]. Thi my of coure lo be obied more direcly ju by differeiig (.3). Referece hould lo be mde o he pper by Koym d Kurow, Kummer formul for he muliple gmm fucio [9] where hey how by differe mehod h (which re vlid for < < ) (5.) log log( ) + γ co log Γ ( ) = co * i 4 = + + ( ) log Γ ( ) (5.3) log log( ) + γ 3 i log Γ ( ) = i * co 3 * * + log + 3 Γ( ) ( ) log Γ ( ) 8 * Γ( ) where Γ ( ) =. I hould however be oed h he muliple gmm fucio Γ * ( ) coidered by Koym d Kurow [9] re o he me hoe rdiiolly employed by Bre [7], Admchi [3] ec. 6. Fourier erie for he lerig Hurwiz ze fucio I Boudjelh [] lo developed he followig Hurwiz ype formul for he lerig Hurwiz ze fucio ς (, ) which he defied for σ >, < by

22 (6.) ( ) e ς (,) = Γ () e + d where, uul, σ = Re( ). Whe = we hve for σ >, ς (,) = ς ( ) [36, p.3]. Boudjelh formul i (6.) ς co(+ ) i(+ ) (,) = Γ( ) i + co (+ ) (+ ) d hold uder he me codiio (.3) bove, mely: (6.3) (σ <, < ; < σ, < ) Thi my be wrie (6.4) ς (,) = Γ( ) co[ / (+ ) ] ( + ) The oio η (, ) i frequely ued ied of ς (, ). Guiller d Sodow [4] proved h for ll comple vlue of d comple z uch h Re ( z) < ½ z ( ) (6.5) ( z) Φ ( z,, ) = ( ) z = + where (6.6) Wih Φ ( z,, ) he Hurwiz-Lerch ze fucio Φ ( z,, ) defied by [36, p.] z = we obi Φ (,,) z = z ( + ) ( ) (6.7) Φ (,, ) = + = ( + ) which w deermied i differe mer i equio (4.4.79) i [6]. Wih = we hve

23 ( ) (6.8) + = ( + ) = + ( ) ( ) = = ς () ( + ) which i he He/Sodow ideiy (ee [7] d [34]). We oe h d e e ( ) = ze z ( + ) ( + ) y u Γ( ) e d = e u du = ( + ) ( + ) d we herefore hve he iegrl repreeio [36, p.] (6.9) ( ) e Φ (,,) z = Γ () e z d Comprig (6.), (6.5) d (6.9) we deduce h ( ) ( ) (6.) Φ (,, ) = = ς (, ) = + ( + ) = ( + ) I pper h hi formul rie he reul of pplyig he Euler erie rformio [8, p.44]. Willim d Zhg [4] lo coidered he lerig Hurwiz ze fucio i 993 (bu heir pper w o refereced by Boudjelh []).Willim d Zhg defied J(,) by ( ) J(,) = ( + ) d hey repored h for σ < co(+ ) i(+ ) (,) = Γ( ) i + co (+ ) (+ ) J (which i here reproduced fer ierig fcor of which pper o be miig i equio (.7) d (3.4) of heir pper [4]). There pper o be ome cofuio i equio (.7) of [4] which e h i i vlid for σ < where equio (3.3) of he me pper e h he requiie codiio i σ <. 3

24 Upo eprio of erm ccordig o he priy of we ee h for σ > ( ) = ( + ) ( + ) (+ ) + = ( + /) ( + ( + )/) d we herefore ee h ς (, ) i reled o he Hurwiz ze fucio by he formul (6.) ς (,) + = ς, ς, He d Pric [5] howed i 96 h he Hurwiz ze fucio could be wrie (6.) ς(, ) = ς(,) ς, + d, by lyic coiuio, hi hold for ll. Wih = / hi become (6.3) + ς, = ς(, ) ς, d hece we hve for σ > (6.4) ς (,) ς(,) ς, + = d (6.5) ς = ς ς (,), (,) Sice ς (, ) c be coiued lyiclly o he whole comple ple ecep for imple pole =, ς (, ) c be coiued lyiclly o become eire fucio d (6.), (6.4) d (6.5) herefore hold i he whole comple ple. We ow muliply (c) by ς = ς ς ( ) (, ) ( ), ( ) (, ) 4

25 d e he limi o obi (6.6) ice lim[( ) ς (, )] = lim[( ) ς (, )] =. I hould be oed h we co uomiclly ubiue = i he formul ς () = ( )() ς becue h equio i oly vlid for Re ( ) > (ecludig = ). Foruely, Hrdy [38, p.6] gve he followig fuciol equio for he lerig ze fucio (6.7) ( ) ς ( ) = Γ ( )i( /) ς( + ) = Γ( )i( /) ς ( + ) d i i hi equio h eble u o eque ς () = ς (). A c be ee from Ayoub pper [6], hi i preciely he fuciol equio for he ze fucio which w fir pouled by Euler my yer before Riem. We hve ς = ς i( / ) () ()lim Uig L Hôpil rule reul i co( / ) = () lim = ς ς () log log We oe h ς ς () = lim[( ) ( )] = lim ( ) ς ( ) = lim lim[( ) ς ( )] 5

26 d uig L Hôpil rule gi give u = log We he hve he well-ow reul ς () = I i iereig o oe h ubiuig (.3) i (6.) give u ς co co ( + ) i i ( + ) (,) = Γ( )i + co ( ) ( ) [ ( ) ]co [ ( ) ]i = Γ( ) i + co ( ) ( ) co(+ ) i(+ ) = Γ( ) i + co (+ ) (+ ) d we hve herefore recovered (6.) i rher righforwrd mer. Leig i (6.) give u ς co(+ ) i(+ ) (, ) = Γ ( ) co + i (+ ) (+ ) d uig (6.) hi i equl o = ( ) ( ) + = + The Euler polyomil E ( ) m my be epreed by (6.8) Em() = ( )( ) = + m (which my be cored wih equio (.4)) d hece we obi he well-ow Fourier erie [4] 6

27 (6.9) (6.9b) m m+ + ( ) Em( ) = m+ ( ) 4( m)! i( ) + m m + ( ) Em ( ) = m ( ) 4(m )! co( ) + 7. Some rigoomeric erie The followig ideiie re recorded by He [6, pp. 3 & 44] for Re ( ) < < (7.) > d (7.b) i( + y) ( ) = coec( ) co y ς, co y+ ς, Γ( ) co( + y) ( ) = coec( ) i y+ ς, i y ς, Γ( ) d we my oe h he ecod ideiy my be obied by differeiig he fir oe wih repec o. y Noe h deil ler. N i he fir ideiy, coec( ) ; hi poi i coidered i more Leig (7.) (7.b) y = d we obi i( ) ( ) = coec( )co ς, ς, Γ( ) co( ) ( ) coec( = )i ς, + ς, Γ( ) or equivlely (7.3) i( ) ( ) = coec ς, ς, 4 Γ( ) 7

28 (7.3b) co( ) ( ) = ec ς, + ς, 4 Γ( ) Leig d p = d he ddig he bove wo equio immediely reul i he well-ow Hurwiz formul for he Fourier epio of he Riem ze fucio ς ( p, ) p co p i (7.4) ς ( p, ) = Γ( p) i + co p p ( ) ( ) where Re ( p ) < d <. Boudjelh [] howed h hi formul lo pplie i he regio Re ( p ) <. I my be oed h whe = hi reduce o Riem fuciol equio for ς ( p). Leig = p we my wrie hi (7.5) co i ς (, ) = Γ ( ) co + i ( ) ( ) Thi my lo be wrie [36, p.89] (7.6) Γ() i i ς (, ) = e L (, ) e L(, ) + ( ) where he periodic (or Lerch) ze fucio L (, ) i defied by e L (, ) = i We ow coider he equio (7.3) i he ce where i poiive ieger N > : we ee h (7.7) (7.7b) N i( ) ( ) N = coec ς N, ς N, N 4( N )! N co( ) ( ) N = ec ς N, ς N, N + 4( N )! Uig he fmilir ideiy [5, p.64] for BN (, ) () ς N = N N (which i lo derived i (7.) below) 8

29 we herefore hve ς N, ς N, = BN BN N I pig we oe h lim Nς ( N, ) = = lim B ( ) N N N Uig he well-ow formul for he Beroulli polyomil [36, p.6] we ge for N B ( ) = ( ) B ( ) N ς N, ς N, ( ) B = N N I imilr mer we ee h for N,, N N N ( ) + ς + ς = N N B Accordigly he bove equio (7.7) my be wrie (7.8) (7.8b) N i( ) ( ) N N = coec ( ) B N N 4 N! N co( ) ( ) N N + = ec ( ) B N N 4 N! Leig N N + d = i (7.8) he give u he well-ow Fourier erie for he odd Beroulli polyomil [4, p.338] (7.9) i B = N +, N =,,,... N + N + () ( ) ( )! N + ( ) d leig N N d = i (7.8b) give u he correpodig erie for he eve Beroulli polyomil (7.9b) co B = N, N =,,... N () N + ( ) ( )! N ( ) 9

30 Referrig o (7.7) we ee h we hve ideermie form whe N N we ep bc o (7.3) d hece i( ) ( ) = coec ς, ς, 4 Γ( ) d coider he limi N. Uig L Hôpil rule we ee h ς, ς, + lim coec ς, ς, lim = co N N ς N, ς = N, + d we herefore obi (7.) N i( ) ( ) ς N, ς = N, N (N )! Applyig he me limiig procedure o (7.3b) give u (7.b) N co( ) N ( ) ( ) ς N, ς = N, N + ( N)! + Wih N = i (7.b) we obi co( ) = ς, + ς, d uig Lerch ideiy (.6) hi become = log Γ + log Γ log( ) Employig Euler reflecio formul we obi = log i log d we hereby obi he fmilir Fourier erie [3, p.4] 3

31 co( ) log i = Wih N = i (7.8) we ee h ( ) = i( ) The Clue fucio Cl ( ) re defied by [36, p.5] N i Cl ( ) = N N co Cl ( ) = N + N + d uig (7.) we hve hereby derived Admchi reul [] (N )! Cl N ( ) = N N, N, ) ( ) (7.) ς ( ) ς ( ( N)! ( ) Cl N ( ),, N + = N + N ) ( ) N (7.b) ς ( ) ς ( which were lo obied i equio (4.3.67) i [4] by eirely differe mehod. We recll He formul [7] for he Hurwiz ze fucio which i vlid for ll C ecep for = ( ) ( ) ς (, u) = ( u ) = + = + d wih hi my be wrie ς (, u) = ( ) ( u+ ) + = I i how i equio (A.3) of [9] h m Bm ( ) = ( ) ( + ) + = m 3

32 A differe proof, uig he Hurwiz-Lerch ze fucio, w recely give by Guiller d Sodow [4]. They lo oed h = d we herefore hve ( ) ( + ) m = for > m =,,,... Bm ( ) = ( ) ( + ) + = m Uig he He ideiy hi immediely give u he well-ow reul (7.) Bm (, ) ( ς m = ) m which we ued bove. Wih p = i (7.3) we ge q i( p/ q) ( ) p p = coec ς, ς, 4 Γ( ) q q d from (..) we hve q p p jp j ς, ς, = 4 Γ( )( q) co i ς, q q j= q q p p jp j ς, ς, 4 ( )( ) i i ς, q q = Γ q q q j= q We he ee h i( p/ q) jp j = i ς, q q q j= q d hi geerlie he formul previouly give by Srivv d Tumur [37]. Leig = d y = i (7.) give u 3

33 i(+ ) ( ) = coec( ) co ς, co + ς, Γ( ) ( ) ( ) 9. Some coecio wih he ie d coie iegrl I pig, we meio h here ei oher rigoomeric epio for ψ ( ),log Γ ( ), logg( + ), ς (, ) ec. Thee re e ou below (furher deil re coied i [8]). (8.) ψ ( ) = log + [co( ) Ci( ) + i( ) i( )] which pper i Nörlud boo [33, p.8]. (8.) log Γ ( ) = log( ) + log + [i( ) Ci( ) co( ) i( )] Cl ( ) log Γ( ) log G( + ) = log (8.3) ( ) [ co( ) Ci( ) + i( ) Si( ) ] + ς ( ) Elizlde [] repored i 985 h for > (8.4) ς (, ) = ς (, )log [co( ) ( ) i( ) ( )] Ci i = where Si() i he ie iegrl fucio defied by [3, p.878] d [, p.3] i Si( ) = d, Si () = We hve he well-ow iegrl from Fourier erie lyi 33

34 i = d d herefore defiig i( ) = Si( ) we hve i i i i( ) = d d = d The coie iegrl Ci( ) i defied i [3, p.878] d [, p.3] co ( ) Ci( ) = γ + log + d = γ + log + ( )! (where, i he fil pr, we hve imply ubiued he Mcluri erie for he iegrd). We lo hve co Ci( ) = d d hi more clerly how he coecio wih i( ). Ci( ) i frequely deiged ci( ) i oher wor uch [3]. We recll he well-ow ideiy (8.5) γ = lim log + = I i iereig o oe h Sodow [35] h dicovered imilr lerig erie (8.6) 4 + log = lim ( ) log + = + ( ) + = ( ) log + = = 34

35 4 = log log = log where, i he l lie, we hve employed (4.4.) from [6]. Sodow formul my lo be obied from Lerch rigoomeric erie epio for he digmm fucio for < < (ee for emple Growll pper [, p.5] d Niele boo [3, p.4]) (8.7) + ψ ( ) i+ co + ( γ + log ) i= i(+ ).log Leig = / we obi + ψ (/ ) + γ + log = ( ) log d, ice [36, p.] ψ (/ ) = γ log, equio (8.6) follow uomiclly. We my lo wrie (8.7) (8.8) i(+ ) + ψ ( ) + co + ( γ + log ) =.log i The bove formul ugge h iegrio my be fruiful employig he iegrl i he ble [3, p.63] i(+ ) i d = + i = p The iegrl ψ ( ) my lo produce iereig reul. Oe could lo employ he ubiuio i (8.7) + y ( y ) log = dy ylog y bu hi h o bee eplored i y deph. 35

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39 Iroducio o he Geerl Theory of Ifiie Procee d of Alyic Fucio; Wih Accou of he Pricipl Trcedel Fucio. Fourh Ed., Cmbridge Uiveriy Pre, Cmbridge, Lodo d New Yor, 963. [4] K.S. Willim d N.-Y. Zhg, Specil vlue of he Lerch ze fucio d he evluio of ceri iegrl. Proc. Amer. Mh. Soc., 9(), (993), hp://mh.crleo.c/~willim/pper/pdf/8.pdf [4] N.-Y. Zhg d K.S. Willim, Some reul o he geerlized Sielje co. Alyi 4, 47-6 (994). Dol F. Coo Elmhur Dudle Rod Mfield Ke TN 7HD dcoo@bopeworld.com 39