The Arakawa-Kaneko Zeta Function

Transcription

1 The Arakawa-Kaeko Zeta Fuctio Marc-Atoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F-0608 Nice Cedex 2 FRANCE Marc-Atoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr This article origially appeared i The Ramauja Joural 22 (200) Abstract We preset a very atural geeralizatio of the Arakawa-Kaeko zeta fuctio itroduced te years ago by T. Arakawa ad M. Kaeko. We give i particular a ew expressio of the special values of this fuctio at itegral poits i terms of modified Bell polyomial. By rewritig Oho s sum formula, we are i a positio to deduce a ew class of relatios betwee Euler sums ad the values of zeta. Mathematical Subject Classificatio (2000) : Primary M4, M35 ; secodary 40-02, Keywords : Poly-Beroulli umbers ; multiple zeta-star values ; Euler sums ; zeta values. Itroductio The Arakawa-Kaeko zeta fuctio has bee itroduced te years ago by T. Arakawa ad M. Kaeko i []. Let us recall that this is the fuctio ξ k defied

2 for ay iteger k by : ξ k (s) = + t s Γ(s) 0 e t Li k( e t ) dt where Li k deotes the k-th polylogarithm Li k (z) = z. The itegral coverges for R(s) > 0 ad the fuctio ξ k cotiues aalytically to a etire fuc- k tio of the whole s-plae. For k =, ξ (s) is othig else tha sζ(s + ) ad for s =, ξ k () = ζ(k + ). I [], Arakawa ad Kaeko have expressed the special values of this fuctio at egative itegers with the help of geeralized Beroulli umbers B (k) called poly-beroulli umbers. Itroduced by M. Kaeko i [5], these umbers are defied by the geeratig fuctio : Li k ( e z ) e z = =0 B (k) z!. I the case where k =, oe fids agai - apart from the sig for B () - the classical Beroulli umbers. Arakawa ad Kaeko also provide i [] (see their corollary 0) a rather complex expressio of the special values of the fuctio at positive itegers i terms of MZV (multiple zeta value) but, very soo afterwards, a simpler represetatio of the values of ξ k at positive itegers i terms of MZSV (multiple zeta-star value) has bee obtaied by Y. Oho. More precisely, trasformig the origial expressio give i [] by meas of a duality theorem, Oho establishes i [7] that : ξ k (m) = = ζ (k,,..., ), k 2... m }{{} 2 m where ζ (k, k 2,..., k m ) refers to the sum: 2 m k k km m Subsequetly, this expressio has foud a importat cotiuatio i [8] where Oho states ad proves his remarkable sum formula :. m k 2 ζ (k m,,..., ) = 2(k )( 2 k )ζ(k), }{{} m=0 m ad it is also the subject of a iterestig commetary i [6]. I this article, we itroduce the more geeral fuctio ξ k (s, x) defied for R(s) > 0 ad x > 0 by : ξ k (s, x) = + e xt Li k( e t ) t s dt Γ(s) 0 e t 2

3 which is a very atural extesio of the Arakawa-Kaeko zeta fuctio i the same way as the Hurwitz zeta fuctio ζ(s, x) geeralizes the Riema zeta fuctio : oe has ξ k (s, ) = ξ k (s) ad, i the case where k =, the fuctio ξ (s, x) is othig else tha the classical sζ(s +, x). Followig the same patter as i [], we show that this fuctio ξ k (s, x) cotiues aalytically i the whole complex s-plae as a etire fuctio of s, ad we express its special values at egative itegral poits by meas of geeralized Beroulli polyomials B (k) (x) whose values at 0 are precisely the poly-beroulli umbers (cf. theorem 2 ad remark 3). I this way, we show that : ξ k ( m, x) = ( ) m B (k) m (x) (m = 0,, 2,... ). Regardig the special values of ξ k (s, x) at positive itegers, we obtai the followig represetatio (cf. theorem ) :! ξ k (m +, x) = ( + ) k x(x + )... (x + ) P m(h () (x),..., h (m) (x)), =0 where P m (x,..., x m ) deotes the m-th modified Bell polyomial defied by the geeratig fuctio : t m exp( x m m ) = P m (x,..., x m ) t m ( ) ad where : m= h (m) (x) = m=0 j=0 (j + x) m. ( ) This Hasse formula exteds the represetatio already give i [3] i the case k = (cf. remark 2). Specializig this expressio at x =, we the deduce a ew expressio for the values of the Arakawa-Kaeko fuctio at positive itegers (cf. corollary ) from which follows the decompositio : ξ k (m + ) = ζ (k,,..., ) = }{{} m P m(h k, H,..., H (m) ) where H, H,..., H (m) deote the harmoic umbers (cf. corollary 2). This leads us to the rewritig of Oho s sum formula i the followig form : k 3 m= P m(h k m, H,..., H (m) ) = [(k 2) (k )2 2 k ]ζ(k), which defies a ew class of relatios betwee Euler sums ad the values of zeta (cf. corollary 3 ad example 3). This class cotais i particular (i the simplest H case where k = 4), the famous relatio = 5 ζ(4) whose origi goes back to 3 4 Euler ad Goldbach (cf. [4]). 3

4 2 A geeralized Arakawa-Kaeko fuctio Propositio. Let be a Laplace-Melli itegral with : F (s, x) = + e xt f(t)t s dt Γ(s) 0 f(t) = a + ( e t ) =0 where the coefficiets a are supposed to satisfy the coditio a = O( ). The followig properties hold : ) The itegral F (s, x) coverges for R(s) > 0 ad x > 0. 2) If m is a atural umber ad s = m + the : F (m +, x) = where P m ad h (m)! a + x(x + )... (x + ) P m(h () (x),..., h (m) (x)) () =0 are respectively give by formulas ( ) ad ( ). Proof. By our assumptio o a, there exists a costat C > 0 ad a iteger N such that for all t 0 : a ( e t ) ( e t ) C C =N =N ( e t ) = Ct e t which esures the covergece of the itegral ad authorizes the iversio of ad : + F (s, x) = a + e xt ( e t ) ts 0 Γ(s) dt. =0 The, formula () results from the followig lemma. Lemma. For x > t, oe has : + 0 e xξ ( e ξ ) e ξt dξ =! x(x + )... (x + ) exp( m= h (m) (x) tm m ). Proof. For a > 0 ad b > 0, let us start from the classical Euler s relatio : B(a, b) = 0 u a ( u) b du = Γ(a)Γ(b) Γ(a + b). 4

5 Puttig : u = e ξ, a = x t ad b = +, oe deduces : + Moreover, oe has : 0 e xξ ( e ξ ) e ξt dξ =! (x t)( + x t)... ( + x t).! (x t)( + x t)... ( + x t) =! x(x + )... (x + ) ( t k + x ) = =! x(x + )... (x + ) exp(! x(x + )... (x + ) exp( = k=0 k=0 m= l( k=0 t k + x )) t m m(x + k) m )! x(x + )... (x + ) exp( m= h (m) (x) tm m ). Applyig ow the previous propositio with a + = (+) k, oe has : f(t) = =0 ( e t ) ( + ) k = Li k( e t ) e t, ad we immediately obtai the followig theorem : Theorem. Let k a iteger. The Laplace-Melli itegral : ξ k (s, x) = + e xt Li k( e t ) t s dt Γ(s) 0 e t coverges for R(s) > 0 ad x > 0. Moreover oe has : ξ k (s, x) = =0 + e xt ( e t ) ts ( + ) k 0 Γ(s) dt. I particular, if m is a atural umber ad s = m +, the : ξ k (m +, x) =! ( + ) k x(x + )... (x + ) P m(h () (x),..., h (m) (x)). =0 Remark. I the case x =, oe has : ξ k (s, ) = + e t Li k( e t ) t s dt = + t s Γ(s) 0 e t Γ(s) 0 e t Li k( e t ) dt = ξ k (s). Thus, i this case, oe fids agai the origial Arakawa-Kaeko zeta fuctio itroduced i []. 5

6 Remark 2. I the case k =, oe has : that : ξ (s, x) = + 0 Li ( e t ) e t = t e t e xt t ( e ) t s 2 dt = (s )ζ(s, x). t Γ(s ) from which follows Thus, i this case, idetity is othig else tha the represetatio for the values of the Hurwitz zeta fuctio (called Hasse formula) give i [3] : (m + )ζ(m + 2, x) = Example. 2ξ k (3, x) =! ( + )x(x + )... (x + ) P m(h () (x),..., h (m) (x)). =0 ξ k (, x) = ξ k (2, x) = =0 =0 =0! ( + ) k x(x + )... (x + ) ;! ( + ) k x(x + )... (x + )! ( + ) k x(x + )... (x + ) [( i + x )2 +! 6ξ k (4, x) = =0 ( + ) k x(x + )... (x + ) [( i + x )3 + 3 (i + x) i + x ; (i + x) (i + x) 2 ] ; (i + x) 3 ]. Theorem 2. For all iteger k et real x > 0, the fuctio s ξ k (s, x) aalytically cotiues to the whole complex s-plae ad its values at egative itegers are give by : ξ k ( m, x) = ( ) m B (k) m (x) (m = 0,, 2,... ) (3) where B (k) (x) is defied by the geeratig fuctio : e xz Li k( e z ) e z = =0 B (k) (x) z!. Proof. We apply the classical method to aalytically cotiue a fuctio defied as a Melli trasform (cf. [2]). Oe splits up ξ k (s, x) as the sum of two itegrals : ξ k (s, x) = e xt Li k( e t ) t s dt + + e xt Li k( e t ) t s dt. Γ(s) 0 e t Γ(s) e t 6

7 The secod itegral coverges absolutely for all s C ad x > 0 ad cacels at egative itegers (because ( m) = 0 for m = 0,, 2,... ). For R(s) > 0, the first Γ itegral may be writte : from which follows that : Γ(s) =0 B (k) (x)! + s, lim ξ k(s, x) = ( lim s m s m )B(k) m (x) Γ(s)(m + s) m! = ( ) m B (k) m (x). Remark 3. From the geeratig fuctio which defie them, the polyomials B (k) (x) are give by : B (k) (x) = q=0( ) q( q ) B q (k) x q where B (k) = B (k) (0) are the poly-beroulli umber itroduced by Kaeko i [5]. I particular, specializig idetity (3) at x =, oe fids agai : ξ k ( m, ) = ( ) m B (k) m () = m q=0( ) q( m q which is othig else tha the expressio give by Arakawa ad Kaeko (cf. [], theorem 6). ) B (k) q 3 New expressio of ξ k (m + ) Specializig idetity at x =, oe obtais the Corollary. For all atural umbers m 0 ad itegers k, oe has : ξ k (m + ) = P m(h k+, H,..., H (m) ) with H (m) = j= j m. (4) Example 2. ξ k () = ζ(k + ) ; ξ k = H k+ ; 7

8 ξ k (4) = 6 [ ξ k (3) = 2 [ (H ) 3 k+ (H ) 2 k H k+ ] ; H H + 2 k+ H (3) k+ ] ; ξ k (5) = 24 [ (H ) k+ (H ) 2 H k+ + 3 (H ) k+ H H (3) + 6 k+ H (4) k+ ]. 4 Rewritig of Oho s sum formula ad applicatio to Euler sums From the compariso of (4) with the expressio give by Oho (cf. [7] theorem 2 ad [6] paragraph 2) : ξ k (m) = ζ (k,,..., ) := }{{} m 2 m oe ca immediatly deduce the followig decompositio : k 2... m, Corollary 2. For all atural umber m 0 ad iteger k 2, ζ (k,,..., ) = }{{} m P m(h k, H Rewritig ow Oho s sum formula (cf. [8], theorem 8) : k 2 ζ (k m,,..., ) = 2(k )( 2 k )ζ(k), }{{} m=0 m..., H (m) ). (5) thaks to the precedig decompositio (5), ad takig ito accout that ξ (k ) + ξ k () = kζ(k), oe obtais the followig formula which defies a ew class of relatios betwee Euler Sums ad the zeta values : Corollary 3. For all itegers k 4, k 3 m= P m(h k m, H,..., H (m) ) = [(k 2) (k )2 2 k ]ζ(k) (6) 8

9 Example 3. H = 5 ζ(4) (Euler ad Goldbach) ; 3 4 H [ (H ) H 3 ] = 5 2 ζ(5) ; H [ (H ) H ] [ (H ) H H H (3) 3 ] = 59 6 ζ(6) ; H [ (H ) 2 H + 5 ] [ (H ) [ (H ) 4 (H ) 2 H (H ) H H H H (3) H (3) 4 ] H (4) 3 ] = 77 6 ζ(7). Ackowledgemets The authors wish to thak Professor Yasuo Oho from Kiki Uiversity for the iterest he has show regardig this work. Refereces [] T. Arakawa, M. Kaeko, Multiple zeta values, Poly-Beroulli umbers ad related zeta fuctios, Nagoya Math. J., 53 (999), [2] P. Cartier, A itroductio to zeta fuctios. I : From Number Theory to Physics, pp. -63, Spriger, Berli, 995. [3] M-A. Coppo, Nouvelles expressios des formules de Hasse et de Hermite pour la foctio zeta d Hurwitz, Expositioes Math. 27 (2009), [4] L. Euler, Meditatioes circa sigulare serierum geus, (775), Opera Omia I-5, [5] M. Kaeko, Poly-Beroulli umbers, J. Th. Nombres de Bordeaux, 9 (997),

10 [6] M. Kaeko, A ote o poly-beroulli umbers ad multiple zeta values. I : Diophatie Aalysis ad Related Fields. AIP Coferece Proceedigs, Volume 976, pp (2008). [7] Y. Oho, A geeralisatio of the duality ad sum formulas o the multiple zeta values, J. Number Theory, 74 (999), [8] Y. Oho, Sum relatios for multiple zeta values. I : Zeta fuctios, Topology, ad Quatum Physics, Dev. Math. 4, 3-44, Spriger, New York,