Euler-Hurwitz series and non-linear Euler sums. Donal F. Connon. 11 March 2008

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1 Euler-urwitz series ad o-liear Euler sus Doal F. Coo March 8 Abstract I this paper we derive two epressios for the urwitz zeta fuctio ς ( q+, ivolvig the coplete Bell polyoials Y (,..., i the restricted case where q is a positive iteger ad q, oe of which is Γ( Γ( ς ( q+, Yq! (,! (,..., ( q! ( q! Γ ( + ( ( ( q ( where ( ( are the geeralised haroic uber fuctios defied by ( (. ( + This i tur gives rise to Euler-urwitz series of the for ( ( ( ( ( Γ( Γ( ς (5, ( + ( ( + ( 4! Γ ( + ad siilar epressios ay the be used to deterie idetities for cobiatios of both liear ad o-liear Euler sus; two such eaples of which are show below ( ( ( ( + + 4! ς (5 ( 4 ( ( ( ( ( ( ( ( ! ς (6 INTRODUCTION The Riea zeta fuctio ς (s is defied for cople values s by [46, p.96] ( ς ( s ( (+ s s s s s, (Re (s >

2 + ( ς a ( s, (Re > ; (s s s s s where ς ( s is the alteratig zeta fuctio. a I their 995 paper Melli Trasfors ad Asyptotics: Fiite Differeces ad Rice s Itegrals, Flajolet ad Sedgewic [9] eployed the followig lea: Let ϕ ( z be aalytic i a doai that cotais the half lie [,. The the differeces ϕ ( adit the itegral represetatio of the sequece { } ( (! ( ϕ( ϕ( z πi C z( z...( z dz where C is a positively orieted closed curve that lies i the doai of aalyticity of ϕ ( z, ecircles [, ad does ot iclude ay of the itegers,,...,. Usig the above lea with ϕ (, Flajolet ad Sedgewic [9] also proved the followig idetity: Defiig S ( by ( ( S( the S ( ca be epressed i ters of the geeralised haroic ubers as (4 ( ( ( S( !!!... where ( are the geeralised haroic ubers defied by ( (5 j j The first few values of S ( give by Flajolet ad Sedgewic are: (6. S ( ( (

3 ( + (6. S ( ( ( ( ( (6. S( ( ( ( ( ( Adachi s 996 paper, O Stirlig Nubers ad Euler Sus [] cotaied the followig idetities: ( (7. ( + ( ( (7. ( + ( ( + ( ( (7. ( ( ( + ( + + ( ( ( ( j j ( j As a atter of iterest, I also foud forula (7. reported by Leveso i a 98 volue of The Aerica Matheatical Mothly [] i a proble cocerig the evaluatio of (8 ( e log d γ ς ( Γ + where γ is Euler s costat defied by (9 γ li( log (ad, regardig the above itegral, refer to (4. below. The third idetity (7. is equal to S ( ad hece we have ( ( S( ( ( ( ( (

4 ( + ( ( The followig forula was foud by Sodow [4] by applyig the Euler series trasforatio ethod (which is covered i Kopp s ecellet boo [8, p.4] to the alteratig Riea zeta fuctio. Sodow s result was ( ( ς a ( s + s ( + where the alteratig Riea zeta fuctio is defied by ς + ( ( a ( s s s ( + ad is soeties called the Dirichlet eta fuctio ad ofte desigated by η ( s. It is ow that ς ( s is a aalytic fuctio for Re ( s >. a The idetity ( has soe history: it was cojectured by Kopp ( aroud 9, the proved by asse [5] i 9 ad subsequetly rediscovered by Sodow i 994. asse ( also showed that ( (. ( ς s s s + ( + ( (. ς (, s s s + ( + where ς (, s for Re ( > is the urwitz zeta fuctio. The above two ( s + s forulae are valid for all s ecept s. It ay be iediately oted that ς ( s, ς ( s. It is show i equatio ( i [4] that log y y (. Lis ( y + Lis( y y ( s s s + ( + ad with y this reverts to (.. A differet proof of (. has recetly bee give by Aore []. 4

5 It is show i equatio ( i [4] that ( ( ς (, s + ( + a s where ς ( su a, ay be regarded as a alteratig urwitz zeta fuctio ad this ay be writte as (. ς (, s a ( ( + s At first sight, the two asse idetities (. ad ( loo rather differet. owever, if we cosider the fuctio defied by ( f(, t s t ( + we see that they are i fact itiately related. Ideed we have ( ( s ς ( s f( t, s s + ( + s d ( ς a ( s f(/, + s ( + s It is also show i equatio (4.4.99aiv i [4] that s yu u ( u e + e t t d s ( ( u y s + Γ u ( + e t A uber of idetities were deteried i [] to [5] by usig the asse forula, a cobiatio of the Flajolet ad Sedgewic forulae (6 ad the Adachi idetities (7. Soe of these idetities are collected below for ease of referece. ( ς a( s + ( + ς a ( log s 5

6 ( ς a ς ( a ( ( {( + } ( ς a (4 ( + 6 ς a (5 ( ( + ( ( ( + { } 4 ( ( 6( ( ( 8 ( ( ( ( ( ς ( + ( s s s + ( + ς ( 4 ( ς (! ( + ς (4 ( +! ς (5 ( ( { } ( ( ( ( ( ( + + 4! { } O the basis of this liited data oe ay cojecture that + ( 4 + (4. (4. ( ς ( s + s s + +, ( ς a ( s + s, s s These cojectures were proved (ad i fact geeralised i equatios ( ad ( respectively of [4]: the geeralised idetities are 6

7 ( (4. s s ( s+ Li ( + log Lis ( + + (4.4 s Li ( s+ where the polylogarith fuctio Li ( is defied by (4.5 Li (, ( We also ote that Spieß [44] has derived the followig idetities (5 + + ( + + ( ( ( ( 4 ( ( ( ( ( ( ( Referece should also be ade to the paper by Larcobe et al. [] where they show that for itegers,. (6. + ( ( (6. ( + (6. ( ( + ( ( 6 4 ( I their paper they eploy itegrals of the type 7

8 p (7 e ( e d. ad, if we use the substitutio equatio (4.4.6 i [] t e, we ca iediately see the relatioship with ( s (7. ( s g ( t t log tdt where (7. g ( is defied by (ad this fuctio is also eployed i (8.5 below! Γ ( + Γ( g ( ( +...( + Γ ( + + We also ote the siilarity with the followig idetity give by Aglesio i [5]. (7. ( e ( d a a r + + ( r! a r e r ( ( log( where a ad r (ecept for a, r. I fact, Aglesio s proof cotais the idetity for a, p> (7.4 a p ( e ( e dγ( p ( a+ p Note that by lettig i (6. ad (6., we obtai equatios (4.4.7 ad (4.4. respectively of [4] ( + ( ( ( + + {( } ( ( ( + I additio, Larcobe et al. also show how the above forulae ca be derived usig the idetity fro Gould s boo, Cobiatorial Idetities [] (8 y+ f ( f( + y y ( y + 8

9 where f (t is a polyoial of degree. I [], Gould attributes the forula (8 to Melza []. Referece should also be ade to the paper by Kirschehofer [7]. Alterative proofs of the Larcobe at al. idetities are give i [4]. Verasere [48] has also cosidered various haroic sus ad related itegrals. It is show i [4] that the geeralised haroic ubers also feature i soe ow logarithic itegrals related to the derivative of the beta fuctio (9. ( t log tdt ( ( ( (9. ( t log tdt + ( (9. ( ( ( ( ( log 6 6 t tdt + + ad it is proved i equatio (4.4.55zi of [5] that (9.4 p+ p ( ( ( t log tdt p! p Ideed, sice ( t log tdt t log ( t tdt, oe would autoatically epect a coectio with the Stirlig ubers of the first id defied below i (.. After that legthy itroductio, we ow eed to recall soe properties of the Stirlig ubers s (, of the first id ad the (epoetial coplete Bell polyoials. These are cosidered i the et two sectios. STIRLING NUMBERS OF TE FIRST KIND The Stirlig ubers fuctio s (, of the first id [46, p.56] are defied by the geeratig (. (...( s(, + ad also by the Maclauri epasio due to Cauchy 9

10 (. log ( +! s(, <! d Sice s (, log ( + it is clear that s (, (as is also! d evidet fro the polyoial epressio i (.. We also have s (,. The followig proof of (. was give by Póyla ad Szegö i [7, p.7]. Fro the bioial theore we have ( t ( + ( +...( + t! Usig (. this becoes ( t s (, (! ( t + s (,! ( ( t + s (,! ( s (, + ( ( t! We the ote that ( t ep[ log( t] log! t ad upo coparig coefficiets of we obtai (.. The first few Stirlig ubers s (, of the first id are give i [4] ad also i the boo by Srivastava ad Choi [46, p.57] (. s (, δ,

11 s + (, ( (! s (, ( (! + (! s (, ( ( { } ( ( ( {( } (! s (,4 ( + 6 The above represetatios should be copared with the idetities foud by Larcobe et al. as set out above i (6. A eleetary ethod for deteriig the Stirlig ubers s (, of the first id is show below i (44.6 et seq. COMPLETE BELL POLYNOMIALS The (epoetial coplete Bell polyoials ay be defied by Y ad for (! (,...,... π ( Y!!...!!!! where the su is tae over all partitios π ( of, i.e. over all sets of itegers that ( The coplete Bell polyoials have iteger coefficiets ad the first si are set out below (Cotet [, p.7] ( Y( j such Y (, + Y (,, + + Y (,,, Y (,,,, Y (,,,,,

12 The total uber of ters π ( icreases rapidly; for eaple, as reported by Bell [6] i 94, we have π ( ters. The odus operadi of the suatio i (. is easily illustrated by the followig eaple: if 4, the is satisfied by the itegers i the followig array 4 4 ad these powers feature i the series below ς a (5 { } 4 ( ( 6( ( ( 8 ( ( ( ( ( The coplete Bell polyoials are also give by the epoetial geeratig fuctio (Cotet [, p.4] j t t t ( ep j + Y(,..., Y(,..., j j!!! Let us ow cosider a fuctio f ( which has a Taylor series epasio aroud : we have j j f ( + t ( j t f ( ( j t e ep f ( e ep f ( j! j! j j We see that ( f ( (, ( (,..., ( e + Y f f f ( ( t! d d t f ( f ( + t f ( + t e e e t t ad we therefore obtai (as oted by Kölbig [] ad Coffey [8]

13 d d f ( f ( ( ( ( (4 e e Y ( f (, f (,..., f ( Differetiatig (4 we see that d Y + f (, f (,..., f ( f ( Y f (, f (,..., f ( d ( ( ( + ( ( ( ( ( + ( As a eaple, lettig f ( log Γ( i (4 we obtai d d log Γ( ( ( ( (4. e Γ ( Γ( Y ( ψ(, ψ (,..., ψ ( t t e log tdt ad sice [46, p.] ψ ( ( p! ς ( p+, ( p p+ ( we ay epress Γ ( i ters of ψ ( ad the urwitz zeta fuctios. I particular, Coffey [8] otes that (4. ( Γ Y γ ( (,,..., p+ ( where p ( p! ς ( p+. Values of Γ ( are reported i [46, p.65] for ad the first three are Γ ( ( γ ( Γ ( ς ( + γ ( Γ ( ς ( γς ( γ We have [46, p.] ψ γ log ψ ς + (!( ( ( + +

14 ( ad therefore we ay readily obtai a epressio for Γ. The first te values are also reported i [46, p.66]. We could also, for eaple, let f ( logsi( π i (4 to obtai coplete Bell polyoials ivolvig the derivatives of cot( π. The followig is etracted fro a series of papers writte by Sowde [4, p.68] i. Let us cosider the fuctio f ( with the followig Maclauri epasio b (4. log f ( b + ad we wish to deterie the coefficiets a such that f ( a By differetiatig (4. ad ultiplyig the two power series, we get a b a Upo eaiatio of this recurrece relatio it is easy to see that a a b b b b +! [,,,...,( ] where the sybol [ a, a, a,..., a ] is defied as the deteriat a a a a 4 ( a a ( ( a a a a a a a a a Sice log f ( log a b we have + b + f( e [ b, b, b,...,( b ]! 4

15 Multiplyig (4. by α it is easily see that α b + (4.4 f ( e + [ αb, αb, αb,...,( αb ]! ad, i particular, with α we obtai b (4.5 e + [ b, b, b,...,( b ] f (! α Differetiatig (4.4 with respect to α would give us a epressio for f ( log f(. We ote the well ow series epasio (which is also derived as a by-product later i equatio (46 ς ( log Γ ( + γ + (, < ad hece we have (4.6 Γ ( + + [ γ, ς(, ς(,..., ς( ]! ad (4.7 + [ γς, (, ς(,..., ς( ] Γ ( +! where ς ( is defied as equal to γ. Differetiatig (4.6 we get (...( + ( [ γ, ς(, ς(,..., ς( ]! ( Γ + ad hece, lettig, we have the th derivative of the gaa fuctio i the for of a deteriat (4.8 ( Γ ( [ ς(, ς(, ς(,..., ς( ] where we agai desigate ς ( γ. 5

16 I his paper The asyptotic behaviour of the Stirlig ubers of the first id [5], Wilf proved that if is the (sigless Stirlig uber of the first id, the for each fied iteger we have (the sigless Stirlig uber of the first id is defied as the absolute value of s (, (4.9 log log log λ + λ λ + O (! (! (! where λ j are the coefficiets i the epasio (4. λ j Γ( j j Sice Γ ( + Γ( equatio (4.7 ay be copared with (4. I particular, usig the recurrece (obtaied fro the logarithic derivative of the ifiite product for Γ( z (4. j λ+ γλ + ( ς( j λj+ ; λ j we obtai λ λ γ (6 λ γ π 4 λ4 γ γπ + ς ( λ5 γ γ π + π ( This aterial is cosidered further i [7]. ς 6

17 Ofte i atheatics we loo for divie ispiratio but we do ot usually epect to obtai it fro a caoised sait. This is ideed the source of the et rear. The ( calculatio of g (, as defied above i (7., effectively ivolves the derivative of a coposite fuctio g( f( t ad the geeral forula for this was discovered by Fracesco Faà di Bruo ( who was declared a Sait by Pope Joh Paul II i St. Peter s Square i Roe i 988 [4]. I [6] di Bruo showed that (5 d! ( ( g( f( t ( ( ( g f t dt b! b!... b! b ( f ( t! b ( f ( t! b ( f ( t! where the su is over all differet solutios i o-egative itegers b,..., b of b+ b b, ad b b. I our case, the coposite fuctio was of the for g( f( t where gt ( / tad f ( t t( t+...( t+ ad we therefore have a eplaatio why the Flajolet ad Sedgewic forulatio of (4 ad the di Bruo forula both ivolve the use of the coplete Bell polyoials. I [45] Gould reided the atheatical couity of the ot well-ow forula for the th derivative of a coposite fuctio f ( z where z is a fuctio of, aely (6. ( ( ( j j D f( z Dz f( z ( z, for! j j This epressio is frequetly easier to hadle tha the di Bruo algorith. This differetiatio algorith was also reported by Gould [] i 97 i a soewhat differet for i the case where z is a fuctio of s (6. ( ( ( j j ( j Ds f( z Dz f( z ( z Ds z, for! j j A iterestig paper o the role of Bell polyoials i itegratio has recetly bee preseted by Collis [9]. We are ow ready to epad upo the wor origially carried out by Coppo [8]. COPPO S FORMULA The followig propositio was derived by Coppo [8] i. For positive itegers q ad cople values of C,,,..., we have 7

18 (7 (! (, q, q ( (...( ϑ where! (7. ( ( ( ( ϑ (,, Y! (,! (,..., (! ( ad where ( ( is the geeralised haroic uber fuctio defied by (8 ( ( ( + ( Flajolet ad Sedgewic [9] refer to ( as the icoplete urwitz zeta fuctios (ad hece the title of this paper. We ote that ( ( ( where ( are the geeralised haroic ubers. This ay be proved as follows: With the defiitios ad S (,,! ( +...( + ( d S (,, S (,,! d we for the Taylor series (9. S (,, t St (,, We ay write! S (,, t ( t( t+...( t+ 8

19 ! ( t/ ( t/( +...( t/( + S (,, ( t/ ( t/( +...( t/( + ad we ay epress this as S (,, t S (,, ep log( t/( + The, eployig the Maclauri epasio for log( t/( +, we obtai t S (,, t S (,, ep ( + ad reversig the order of suatio this becoes ( t S (,, t S (,, ep ( (9. ( t S (,, ep (! (! We ow recall the defiitio of the coplete Bell polyoials ( t t ep Y(,...,!! ad see that t ( ( ( ( ( ep ( Y! (,! (,...,(! ( Usig (9. above t! S (,, t St (,, 9

20 ad equatig coefficiets of t we see that! (9. ( ( ( ( S (,, S (,, Y! (,! (,...,(! ( We have the well-ow forula (see for eaple [] ad [4]! ( ( +...( + + which ay be writte as ( S ( +,, + + ad differetiatig this q ties gives us (9.4 ( S ( +, q, + ( + q The, equatig (9. ad (9.4 gives us (7. Sice! Γ ( + Γ( ( +...( + Γ ( + + we ay write Coppo s forula as ( ( Γ ( + Γ( Y q q + + q + ( + Γ ( + + ( q! ( ( ( ( q! (,! (,..., (! ( ( ( p p ( p+ Desigatig h ( + ( we see that h ( ( p! + ( ad we ca therefore write ( as ( Γ ( + Γ( ( + Γ ( + + ( q! ( q ( q (. Yq ( h(, h (,..., ( h ( q Coffey [8] also reported a versio of this forula i 6.

21 AN APPLICATION OF TE ASSE IDENTITY FOR TE URWITZ ZETA FUNCTION As etioed above, asse ( showed that ( (. ( ς s s s + ( + ( (. ς (, s s s + ( + where ς (, s for Re ( > is the urwitz zeta fuctio. The above two ( s + s forulae are valid for all s ecept s. It ay be oted that ς ( s, ς ( s. Usig ( ad the asse idetity (. we obtai for s q+ ad q ( Γ ( + Γ( ς ( q+, Yq! + (,! + (,..., ( q! + ( q! + Γ ( + + ( ( ( q ( Γ( Γ( Y q q q! Γ ( + ( ( ( ( q! (,! (,..., (! ( Particular cases of Coppo s forula are set out below. (. ( Γ ( + Γ( Γ ( + Γ( Y ( + Γ ( + + Γ ( + + (. ( ( Γ ( + Γ( ( Γ ( + Γ( ( Y! + ( + ( ( + Γ ( + + Γ ( + + ( Γ ( + Γ( Y ( + Γ ( + + (! ( ( + (,! + ( Γ ( + Γ( (. ( ( ( + ( + + ( Γ ( + + ( Γ ( + Γ( Y 4 ( + Γ ( ( ( ( ( + (, + (, + (

22 Γ ( + Γ( (.4 ( ( ( ( ( ( ( + ( + + ( We have for q Γ ( (4 Γ ( + Γ( Γ( Γ( ς (, + Γ ( + + Γ ( + ad with this siply becoes the Riea zeta fuctio ς ( ς(, ς( ( + Sice ς (, ψ ( we also have (5 ψ ( Γ ( + Γ( + Γ ( + + which is reiiscet of the well-ow forula (6 ψ ( + a ψ ( a + ( (...( + a( a+...( a+ which coverges for Re ( + a >. Accordig to Raia ad Ladda [8], this suatio forula is due to Nörlud (see [5], [6] ad also Rube s ote [4]. I equatio (4..5 i [] it was show that ψ ( (! ( +...( + (6. Γ( Γ( Γ ( + This result was also reported by Rube [4] i 976. We see that (6. is equivalet to equatio (4. This correspodece is cosidered further i (45 below. We have Legedre s duplicatio forula [46, p.7] for t > π (6. Γ( t Γ t+ Γ( t t

23 Lettig t + we obtai ad we see that π Γ (+ π (+! Γ Γ +! ( Γ ( + Γ (! ( +! Γ ece, lettig / i (4 we get It is well ow that ad hece we obtai (7 ς ( (! ς, + ( +! ς s, s ( ς ( s + [! ] + (+! + This has soe structural siilarities to Raauja s forula for Catala s costat G (see equatio (8.5a i [6], Adachi s paper [] ad [6] (8 [ ] [ ] π π (! G 4 ( + 4 (+ (! but the coectio is ore obvious with the epressio cotaied i Raauja s Noteboos (Berdt [6i, Part I, p.89] (8. + [!] [!] G (+ (! 4 (+ (+! I a persoal couicatio Coffey etioed that it sees that equatios (8 ad (8. have to be equivalet due to the duplicatio forula (6. for the gaa fuctio.

24 About four years ago, the author derived the followig epressio for G (6. of [6] (see equatio ta (! G d ( (+! ad it is clear that this is equivalet to (8. provided oe ca prove that ( + Usig (7 ad (8. we ay write ( ς [!] G ( ( (! ( ( which ay possibly be of assistace i deteriig whether the differece ratioal or irratioal. We have fro (4 ς (, ad we therefore obtai (8. ς (, I particular we see that Γ( Γ( Γ ( + B (, t ( t dt ( t log( t dt t, 4 log( t ς dt t 4 ( t I a scitilla teporis, the Wolfra Itegrator iraculously tells us that ( ς 4G is 4

25 ( t ( ( ( ( 4 log 4 log 4 4 log( t 4 dt Li t + t t Li t t ( ( ( ( ( ( + ilog + i t log t + ili i t ilog i t log t ili i t (how could a hua beig deduce that?. We the have the defiite itegral log( t ( ( ( ( 4 dt Li Li + i Li i Li i 4 t ( t ad referece to the defiitio of the dilogarith fuctio readily shows us that ( ( Li i Li i ig ad we therefore deduce that (8. ς ψ π + 4 4, 8 G as previously derived by Kölbig [i] i a uch sipler aer. I passig, it ay be oted that whilst the beta fuctio B( uv, [4] is oly defied for Re ( u > ad Re ( v >, we ay still deterie Bu (, v as follows: u Sice (, ( v we have B uv t t dt u (, log v B uv t ( t ( t dt v v u ad Γ( u Γ( v Buv (, also iplies that Γ ( u + v B( uv, Buv (, ( v ( u v v ub( u, v [ ψ ψ + ] [ ψ( u+ v ψ( v ] u 5

26 Γ ( u+ Γ( v ad sice ub( u, v Γ ( u+ v we see that [ ψ( u+ v ψ( v ] Γ ( u+ Γ( v li Buv (, li li u v u Γ ( u+ v u u Therefore we obtai ψ ( v (8.4 ψ ( v v ( t log( t t dt i agreeet with (8.. Equatio (8.4 ay also be derived by differetiatig the wellow itegral [56, p.5] for the digaa fuctio ψ ( v γ + v t t dt Fro the defiitio of the urwitz zeta fuctio we see that ς, ς, (4+ (4+ 6 6G (4+ (4+ For coveiece we ow desigate g ( as (8.5 Γ ( + Γ( g ( Γ ( + + ad differetiatio gives us [ ψ ψ ] g ( g( ( ( + + Therefore differetiatio of (4 results i Γ ( + Γ( ς (, ς(, ψ( ψ( Γ ( + + [ ] 6

27 ad sice [4, p.] (8.6 ( ψ ( + + ψ ( ( + + we obtai (9 Γ ( + Γ( Γ( Γ( ς (, ( ( ( ( + + Γ ( + + Γ ( + B (, ( ( With i (9 this becoes the very failiar su origially derived by Euler ς ( ( ( + ( + We ote that g ( g( ( ( + Differetiatio of (9 results i Γ( Γ( (4! ς (4, ( ( ( ( + ( Γ ( + ad with we obtai a well-ow result (4 ( ( + ς (4! which was also obtaied i a differet way i equatio (4..4 of []. Differetiatio of (4 gives us a ore cople suatio Γ( Γ( (4 ς (5, ( ( ( ( ( ( ( ( + + ( 4! Γ ( + With we get aother ow result 7

28 (4 ( ( ( ( + + ς (5 4! which was also previously derived i equatio (4..47a of [] (ad see the detailed refereces cotaied therei. A further differetiatio gives us (4. ( ( ( ( ( ( (4 Γ( Γ( ς (6, ( ( + ( + ( ( + ( 5! Γ ( + 4 ( ( ( ( ( ( ( ( ( ( ( Γ( Γ( ! Γ ( + ad hece we have a further series ivolvig o-liear Euler sus (4. (6 ( 6 ( ( ( 5 ( ( (4 ( 4 ς ! which we ay copare with (45. below, This differetiatio procedure could obviously be eteded to ifiity ad beyod! Istead, we ow cosider the asse/coppo forula for q. This gives us ς Γ( Γ( ( (, (! Γ ( + ad this siply repeats (9 above (as ideed will also happe for q etc. due to the fact that (. is the derivative of (. ad so o. We ote fro (9. that ( t S (,, t S (,, ep (! (! ad by defiitio we have S (,,! ( +...( + 8

29 We have! Γ ( + Γ( ( +...( + ( + Γ ( + + ad therefore we see that ( + Γ ( + Γ( Γ ( + Γ( S (,, Γ ( + + Γ ( + We ay therefore write (9. as Γ ( + Γ( t Γ ( + Γ( ep ( t ( Γ ( + t Γ ( + or equivaletly Γ ( + Γ( t t t ( log ( Γ( Γ ( + Let us ow cosider the case where t t ad. This gives us Γ ( + Γ ( + t ( log Γ ( + + t t ( It is easily see that Γ ( ( ( ( log + Γ + t log Γ Γ + t Γ ( + + t ( + t Γ ( + t ( ( t t log Γ Γ + log + Γ ( + t ad we therefore have Γ( Γ ( + t t ( log log + + Γ ( + t t ( ( t ( + t ( 9

30 ( ( t We the ed up with (44 Γ( Γ ( + t ( log Γ ( + t ( t DETERMINATION OF TE STIRLING NUMBERS s (, Soe tie ago the author cosidered a siilar epressio to (44 i []. Let The we have f ( log Γ ( + log Γ ( + f ( ψ ( ψ ( ( (! ( ( ( + ( ad we therefore have the Maclauri epasio equivalet to (44 (44. + ( ( f ( log Γ ( + log Γ ( + log Γ ( + This ay be writte as (44. + Γ ( + ( ( ep ( ( Γ + Γ ad epasio of the epoetial fuctio leads to (44. ( ( Γ ( + ( ( ( ( ( ( ( O( Γ ( + Γ( 6 ad (44.4 ( ( Γ ( + Γ( ( ( ( ( ( ( ( O( Γ ( + 6

31 The ters ivolvig the geeralised haroic ubers are truly ubiquitous! I cae across (44. i a 999 paper etitled Aalytic two-loop results for self eergy- ad verte-type diagras with oe o-zero ass by Fleisher et al []. Lettig i (44. results i the failiar Maclauri epasio for log( +. Fro (44. ad usig the forulae (. for the Stirlig ubers of the first id we see that + Γ ( + Γ ( + s s s s O 4 ( (, (, + (, (,4 + ( ad this ay be writte usig the Pochhaer sybol ( as (44.5 Γ ( + Γ ( + + ( ( s (, Γ( Γ ( + which is siply the geeratig fuctio for the Stirlig ubers of the first id (lettig i (. + + ( ( +...( + ( s (, ( s (, sice s (, for +. Usig (44.5 is probably the siplest way of evaluatig the Stirlig ubers (by successive differetiatio; for eaple we have Γ ( + ψ( + + ψ( + ( s(, Γ ( + + (44.6 [ ] ad with we obtai + (, ( (! s. A further differetiatio results i Γ ( + [ ( ( ][ ( ( ( ψ + ψ + ψ + + ψ + Γ + Γ ( Γ ( + + ( [ ψ ( ψ ( ] [ ψ( ψ( ] ( ( s(, ] ad with we easily obtai ( s (, ( (!. These results ay be geeralised as follows. The Pochhaer sybol ay be epressed as

32 ( u+ ( u+ ( + u+...( + u+ ad we see that u( + u( + u...( + u u + u + u ( u u + u + u ep log + u + u + u j j+ u ep ( j ( j+ u ep ( ( j+ u j We therefore obtai u ( ep ( ( ( u+ ( ep ( ( u ( u ad we ow that + ( u+ ( s(, ( u+ Differetiatio results i + ( s (, u j j j j + ( s (, u j j j j j

33 d d ad we see that + ( u+ ( s(, u j( j..( j r+ j j j r r j j r r d d r ( u+ ( s(, r! u r r + r We also have fro (4 d u e Y f f f d r + ( f ( ( ( ( r ep ( ( r (, (,..., ( r ( where u ad we have + ( ( ( ( f With f ( ( (...( p ( u p ( p + ( + we see that f ( ( ( p! ( u ( p p ( p ad we thereby obtai d u u Y u u r u d r + ( ( ( r ( r ep ( ( ( r (,! (,...,( (! ( r ( This the gives us + r ( ( r ( r (44.7 r! ( s(, u ( u Yr( ( u,! ( u,...,( ( r! ( u r r ad i particular, with u, we obtai usig (!! r! + (44.8 ( (, ( (,! ( r,...,( (! ( r s Y r r r r Sice ( + ( + we ca show that

34 + r + (44.9 ( s ( +, r+ ( s (, r r ad hece, as reported by Cotet [], we have i ters of the coplete Bell polyoials! r! + r (44. ( ( ( r (, (,!,...,( (! ( r s+ r+ Y r r I subsequetly discovered that this is a slightly differet versio of the proof origially give by Kölbig [] i his 99 thesis. FURTER EXAMPLES OF EULER-URWITZ SUMS Eployig the Nörlud represetatio for the digaa fuctio (see equatio (6 above, it was also show i equatio (4..48 i [] that (45 ψ ( q ( q! + ( sq (, ( +...( + ad with we obtai (45. ψ ( q + ( ( q! s(, q.! Sice [46, p.] ψ ( ( q! ς ( q+, we have ( q q+ (45. p ( ς ( p + ( s (, p.! This result was previously obtaied i 995 by She [4] by eployig a differet ethod. Other proofs were recetly give i [7] ad [46, p.5]. We ay therefore write (45 as ς ( q+, ( q+ + ( sq (, ( +...( + ( q+ + ( sq (, ( + Γ( Γ ( + + 4

35 ad therefore we obtai (45. ς ( q+, ( q+ + ( sq (, Γ( Γ ( + We ay write (44. as (! sq (, ( Y,!,...,( ( q! ( q! ad substitutig this i (45. gives us ( ( ( ( + q q q q ( Γ( (! ς ( q+, ( ( Y,!,...,( ( q! + q+ + q ( ( q ( q Γ ( + ( q! This ay be slightly siplified to Γ( Γ( ( q! Γ ( + q ( ( ( q ( q (45.4 ς ( q+, Yq (,!,...,( ( q! where we particularly ote that i this epressio the coplete Bell polyoials do ot cotai the variable (ad is therefore a copletely differet represetatio fro ( which is reproduced below for ease of copariso. Γ( Γ( ς ( q+, Yq! (,! (,..., ( q! ( q! Γ ( + ( ( ( q ( With q i (45.4 we get Γ( Γ( Γ( Γ( ς (, Y Γ ( + Γ ( + ad we have see this before i (4. Lettig q gives us Γ( Γ( Γ( Γ( ς (, Y ( ( ( Γ ( + Γ ( + Γ( Γ( ( Γ ( + 5

36 We the have (45.5 Γ( Γ( Γ( Γ( ς (, ( Γ ( + Γ ( + ad this ay be cotrasted with (9 ς Γ( Γ( ( (, ( Γ ( + Lettig i (45.5 reproduces the Euler su ( ς ( ad with / we obtai usig (6. ( ( ( ( ( Γ Γ (45.6 ς, Γ ( ( Γ Γ Γ ( where (. + Differetiatio of (45.5 results i ς(4, [ ψ( + ψ( ] [ ψ( + ( ] ( Γ( Γ( Γ( Γ( Γ ( + Γ ( + ψ Usig (8.6 we see that ψ ψ ( ( + ( ( We therefore have or (45.7 ( Γ Γ ( Γ Γ ς (4, ς (4, ( ( ( ( ( ( ( Γ ( + Γ ( + Γ Γ( ( ( ( ( Γ ( + 6

37 Lettig we get ( ( ς (4 ad this cocurs with the followig well-ow results (see, for eaple, equatio (4..46b i [] ( 5 ς (4 4 ( 7 ς (4 4 It is ow fro [4, p.79] that ( ( + ad therefore (45.7 ay be writte as ( ( ( + ( ς (4, Γ Γ( Γ ( + Differetiatio of (45.7 results i ( ( ( ( ( Γ( Γ( ( Γ( Γ( ς (5, + Γ ( + Γ ( + which ay be writte as ( ( ( ( ( + ( (45.7 ς (5, Γ( Γ( 4! Γ ( + We the have with ( + ς (5 ( ( ( 7

38 ( ( ( ( ( + + ad therefore (45.8 ( ( ( ( ( ς (5 + ad this cocurs with the four well-ow idividual Euler sus (which are also derived i []. Differetiatig (45.7 gives us ( ( ( ( ( + ( (45.9 ς ad with we obtai Γ( Γ( ( (6, ( 6 Γ ( + ( ( ( ( ( ( ( + ( Γ( Γ( + 6 Γ ( + (45. ( 4 ( ( ( ( ( ( ( ( ! ς (6 which ay be writte as 5! ς (6 ( ( ( ( ( ( ( + + Four other represetatios of ς (6 ivolvig the geeralised haroic ubers are derived i equatio (4..65 i []. It is possible that this set of siultaeous liear equatios, i cojuctio with Zheg s recet paper [5] ad other ow results, ay be sufficiet to evaluate ore cople o-liear Euler sus ad perhaps verify Coffey s cojecture [7i] that ( ς (6 + ς ( ( + 4 8

39 which we ay epress i stadard for by writig. + ( ( + A CONNECTION WIT TE GAUSS REPRESENTATION OF TE GAMMA FUNCTION We recall (44. ad, as oted by Wilf [5], we have + Γ ( + ( ( ep Γ ( + Γ( + ( ( ( ep ep + ( ( ( ep ( log ep( log ep We therefore have + ( ( Γ ( + ep ( Γ ( + Γ( ep ( log ad, usig (7., this ay be epressed as + ( ( ( +...( + ep ( +! ep ( log We ote fro (9 that ad we see that ( ( log li ep e γ ( ( li ep ep ( + + ( ς We have the Gauss epressio [46, p.] for the gaa fuctio 9

40 ! Γ ( li ( +...( + ad hece we obtai ( ep ς ( e Γ ( + + γ which is equivalet to the well ow series epasio (46 + ( log Γ ( + γ+ ς( With regard to (4 we let f ( log( u+ log( j + u+ j ad therefore we see that f ( ( ( p! ( ( p! ( u+ ( p p p ( p p j ( j+ u+ ece fro (4 we get as before d d ( ( ( ( ( u+ ( u+ Y ( u+,! ( u+,...,( (! ( u+ or equivaletly by referece to (44.5 d Γ ( + u+ Γ ( + u+ Y u + u + u d Γ ( u+ Γ ( u+ ( ( ( (,! (,...,( (! ( ( + SOME INTEGRALS We have Γ ( + Γ( Γ( Γ( B(, Γ ( + Γ ( + 4

41 It is a eercise i Whittaer & Watso [49, p.6] to show that (47 + ( v ( v log B(, log + dv, > ( v log v ad this forula is attributed to Euler (see also [4, p.87]. I passig, we ote the required syetry i ad. Therefore we have usig (44. (47. ad + ( ( ( v ( v log ( v log v + + dv Γ ( + ( v ( v log log + + dv Γ ( + Γ( ( v log v A alterative proof is show below. logv log v We see that v e ad therefore we have! ( v ( v ( v log v dv dv ( v log v! v j log dv! j v v We have j a vv dv ad therefore differetiatio with respect to a results i j + a + j a vv ( (! log vdv ( j+ a+ ece we have j v ( (! log vdv ( j + j v vdv j ( log ( (! 4

42 ad thus (47. + ( v ( v ( ( dv ( v log v + ( ( + ( ( + + ( + ( ( log + Differetiatig (47. results i v ( v dv ( v + ( ad as we have the well-ow i itegral ( v dv v ( Successive differetiatios give us p v ( v log v dv p v + ( ( ( (...( p p+ ad hece with we see that (47. p ( v log v dv p ( p! v ( p+ We also have fro equatio (E.5c i [7] (47.4 p ( v v log v dv p ( ( p p! (! ( p p p+ + v ( + ad this ay also be derived by differetiatig (8.4 4

43 ( t log( t t dt ψ ( Copletig the suatio of (47.4 gives us for u < We therefore obtai ( p+ p p ( ( v v log v ( p! u u v dv (47.5 ( p+ p p ( [log( uv log( u] v log v ( p! u d v v With p we have (47.6 ( ( [log( uv log( u] v log v u d v v ad with we have (47.7 ( [log( uv log( u]log v u d v v The Wolfra Itegrator is oly able to evaluate (47.7 i ters ivolvig the dilogarith ad the trilogarith i the cases where p,. We ote fro equatio (.6f i [] that i ters of the polylogarith fuctio (4.5 ( u u Li u Itegratig (47.5 will give us a epressio for ( p+ u. Usig (47.4 we ay ae a ore geeral suatio for Re ( s > ( p+ p ( ( log s s v p u v v v ( p! u d v ad therefore we get 4

44 (47.6 ( p+ p ( [ Lis( u Lis( uv ] v log v s v p ( p! u d v We have a siilar relatioship fro (4.4.56c i [5] ( u s s u s u v Li ( u Li ( v d v With u i (47.6 we obtai (47.7 ( p+ ( [ ς s ] s p ( p! p ( s Li ( v v log v dv v ad with (47.8 ( p+ [ ς s ] s p ( p! p ( s Li (log v v dv v I (9 we saw that B (, ( ς (, ( where B( uv, is Euler s beta fuctio [4] defied for Re ( u > ad Re ( v > u ( v ad B( uv, t t dt Γ( u Γ( v Buv (, Γ ( u + v Fro (47.4 we have ( ( v v ( dv v ad we therefore obtai a triple itegral represetatio for ς (, ( v v ς (, u du t ( t dt dv v 44

45 The geoetric series the gives us ( tv v ς (, dt du dv v u( t uv( t We have v log[ v( t] du uv( t t ad hece we obtai the double itegral represetatio ( ( tv log[ v( t] log t ς (, ( t( v With we have dtdv log[ v( t] log t ς ( dtdv ( t( v AN INTEGRAL INVOLVING TE URWITZ-LERC ZETA FUNCTION Differetiatio of (47. gives us (48 v ( v ψ ( + ψ ( + dv + v ad with / we have v( v ψ + ψ dv + v or equivaletly v( v ψ + + γ + log + v dv We obtai the suatio v( v + ( γ + log (+ (+ (+ (+ v ψ + dv 45

46 Sice ς ( s s s we have ( + ς ( 4 ( + 7 ς ( 8 ( + ad hece we have ψ + 7 v 7 v + ( γ + log ς( ς( ς( (+ 4 4 v 8 (+ Sice dv v v Φ v,, (+ 4 ( + / 4 where Φ ( vsa,, is the urwitz-lerch zeta fuctio [46, p.] defied by ( vsa,, Φ we ay write this as v ( + a s ψ + 7 v 7 + ( γ + log ς( ς( ς( Φ v,, dv (+ 4 4 v 8 4 I a recet preprit [i], etitled Deteriatio of soe geeralised Euler sus ivolvig the digaa fuctio, usig Kuer s Fourier series epasio for logγ ( it is show that ψ + (48. [ γπ + 7 ς (] ( + 8 ad we therefore obtai the itegral (48. v 7 ς ( Φ v,, dv ς( log ς( v It is also show i [i] that 46

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51 [48] J.A.M. Verasere, aroic sus, Melli trasfors ad itegrals. It. J. Mod. Phys. A4 ( [49] E.T. Whittaer ad G.N. Watso, A Course of Moder Aalysis: A Itroductio to the Geeral Theory of Ifiite Processes ad of Aalytic Fuctios; With a Accout of the Pricipal Trascedetal Fuctios. Fourth Ed., Cabridge Uiversity Press, Cabridge, Lodo ad New Yor, 96. [5].S. Wilf, The asyptotic behaviour of the Stirlig ubers of the first id. Joural of Cobiatorial Theory Series A, 64, 44-49, [5] De-Yi Zheg, Further suatio forulae related to geeralized haroic ubers. J. Math. Aal. Appl. 5 ( Doal F. Coo Elhurst Dudle Road Matfield, Ket TN 7D dcoo@btopeworld.co 5