Series acceleration formulas for beta values

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1 Discrete Mthemtics d Theoreticl Computer Sciece DMTCS vol. :, 00, 3 36 Series ccelertio formuls for bet vlues Kh. Hessmi Pilehrood d T. Hessmi Pilehrood Mthemtics Deprtmet, Fculty of Bsic Scieces, Shhreord Uiversity, Shhreord, P.O. Box 5, Ir. received Sep. 3, 009, revised Mrch 7, 00, ccepted Mrch 3, 00. We prove geertig fuctio idetities producig fst coverget series for the sequeces β +, β + d β+3, where β is Dirichlet s bet fuctio. I prticulr, we obti ew ccelerted series for Ctl s costt coverget t geometric rte with rtio 0, which c be cosidered s log of Amdeberh-Zeilberger s series for ζ3. Keywords: Dirichlet s bet fuctio, Ctl s costt, Apéry-lie series, geertig fuctio, covergece ccelertio, Mrov-Wilf-Zeilberger method, Mrov-WZ pir. I hoor of P. Fljolet s 60th birthdy Itroductio I this pper we cotiue our study o fidig geerlized idetities [8, 9] tht produce fst coverget series for some clssicl costts. I 978, R. Apéry used the fst coverget series ζ3 5 3 first obtied by A. A. Mrov [4] to derive the irrtiolity of ζ3 [7]. This series coverges t geometric rte with rtio /4. A geerl formul givig logous series for ll ζ + 3, 0, ws proved by Koecher [] d idepedetly i expded form by Leshchier []. For <, it hs the form ζ m m. Expdig the right-hd side of by powers of d comprig coefficiets of o both sides leds to the Apéry-lie series for ζ + 3. I prticulr, comprig costt terms 0 recovers c 00 Discrete Mthemtics d Theoreticl Computer Sciece DMTCS, Ncy, Frce

2 4 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood Mrov s formul d comprig coefficiets of d 4 gives the followig two formuls: + ζ ζ7 7 5 j j j, j m m m j respectively. I 996, ispired by this result, J. Borwei d D. Brdley [4] pplied extesive computer serches o the bse of iteger reltios lgorithms looig for dditiol zet idetities of this sort. This led to the discovery of the ew idetity ζ j j, j 4, 3 which is simpler th Koecher s formul for ζ7, d similr idetities for ζ9, ζ, ζ3, etc. This llowed them to cojecture tht certi of these idetities, mely those for ζ4 + 3 re give by the followig geertig fuctio formul [3]: ζ m m m 4 4, <. 4 The vlidity of 4 ws proved lter by G. Almvist d A. Grville [] i 999. Expdig the right-hd side of 4 i powers of 4 gives formul for ζ4 + 3, which for cotis fewer summtios the the correspodig formul geerted by. I prticulr, comprig costt terms gives d comprig coefficiets of 4 yields 3. There exists bivrite uifyig formul for idetities d 4 4 x y x 4 x y 4 m m x + 4y 4 m 4 x m y 4. 5 It ws origilly cojectured by H. Cohe d the proved by D. Brdley [7] d, idepedetly, by T. Rivol [8]. Their proof cosists of reductio of 5 to fiite o-trivil combitoril idetity which c be proved o the bsis of Almvist d Grville s wor []. Aother proof of 5 bsed o pplictio of WZ-pirs ws give by the uthors i [9]. Sice + m 4 x y 4 m0 ζ + 4m + 3x y 4m, x + y 4 <, 6 the formul 5 geertes Apéry-lie series for ll ζ + 4m + 3,, m 0, coverget t the geometric rte with rtio /4 d cotis, s prticulr cses, both idetities d 4. I [9], the uthors showed tht the geertig fuctio 6 lso hs much more rpidly coverget represettio, mely 4 x y 4 r m m x + 4y 4 m m4 x m y 4, 7

3 Series ccelertio formuls for bet vlues 5 where r x x 3 + x 4 8x 5y 4 + 0y 4 + y 4 x. The idetity 7 produces ccelerted series for ll ζ + 4m + 3,, m, coverget t the geometric rte with rtio 0. I prticulr, if x y 0 we get Amdeberh-Zeilberger s series [] for ζ3, ζ It is worth poitig out tht both idetities 5 d 7 were proved i [9] by usig the sme Mrov-WZ pir see lso [0, p. 3] for the explicit expressio, but with the help of differet summtio formuls. A more geerl form of the bivrite idetity 5 for the geertig fuctio m + A 0 ζ + 4m B 0 ζ + 4m C 0 ζ + 4m + x y 4m m0 A 0 + B 0 + C 0 4 x y 4, x + y 4 <, where A 0, B 0, C 0 re rbitrry complex umbers, ws proved i [9] by mes of the Mrov-Wilf- Zelberger theory. More precisely, oe hs where A 0 + B 0 + C 0 4 x y 4 d B 0 5 x m x + 4y 4 m d m m4 x m y 4, L x + 43x 4 + 0y 4 L 45 x d L is solutio of certi secod order lier differece equtio with polyomil coefficiets i d x, y with the iitil vlues L 0 C 0, L 5 x A 0 /5 + 5x 4x 4 + 6y 4 C 0 /30. If we te A 0 C 0 0, B 0 i 9, the L 0 for ll 0 d we get the bivrite idetity 5. If B 0 C 0 0, A 0, the we obti m + ζ + 4m + 4x y 4m m L x x 4 + 0y 4 L 5 x, m m4 x m y 4 where the sequece L is defied recursively s bove. A similr formul is vlid for the sequece ζ + 4m +.

4 6 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood First results relted to geertig fuctio idetities for eve zet vlues belog to Leshchier [] who proved i expded form tht for <, + ζ + Comprig costt terms o both sides of 0 yields Sice ζ π ζ! B, m where B Q re the Beroulli umbers geerted by the expoetil geertig fuctio x e x x B!, m. 0 formul 0 gives Apéry-lie series for eve powers π,,,.... I 006, D. Biley, J. Borwei d D. Brdley [5] proved other idetity ζ + 3 m m 4 m It geertes similr Apéry-lie series for the umbers ζ +, which re ot covered by Leshchier s result 0. I the sme pper [5], geertig fuctio producig fst coverget series for the sequece π +4, 0,,,..., ws foud, which for <, hs the form 3 3 B π ! Here the left-hd side of is Mcluri expsio of the fuctio Comprig costt terms i implies tht π cscπ + 3 cosπ/ ζ m. m. The idetity gives formul for π +4 which for 0 ivolves fewer summtios the the correspodig formul geerted by 0.

5 Series ccelertio formuls for bet vlues 7 I this pper, we cosider vlues of Dirichlet s bet fuctio, which for Rs > 0 is defied by the series βs + s. It is well-ow due to Euler tht for odd s, βs is rtiol multiple of π s β + E +! π+, 0,,,..., where the iteger coefficiets E re the eve idexed Euler umbers defied by the expoetil geertig fuctio coshz ez e z + E! z z! + 5z4 6z ! 6! For eve s, o Euler-type formul is ow, d β defies the well-ow Ctl s costt G : I Sectios, 3 we prove geertig fuctio idetities producig fst coverget series for the sequeces β + d β +. I prticulr, we estblish some logs of idetities 0 d for the odd powers of π : π +, 0, d π +3, 0. As cosequece of our results o eve bet vlues, we derive the followig ice formul for Ctl s costt: where G q , 3 q It c be cosidered s log of Amdeberh-Zeilberger s series 8 for ζ3, sice the series o the right-hd side of 3 coverges expoetilly with rtio 0. Apéry-lie series for β + d β + 3. We strt by recllig severl defiitios d ow fcts relted to the Mrov-Wilf-Zeilberger theory see [4, 5, 6]. A fuctio H,, i the iteger vribles d, is clled hypergeometric or closed form CF if the quotiets H +, H, d H, + H,

6 8 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood re both rtiol fuctios of d. A hypergeometric fuctio tht c be writte s rtio of products of fctorils is clled pure-hypergeometric. A pir of CF fuctios F, d G, is clled WZpir if F +, F, G, + G,. 4 A P-recursive fuctio is fuctio tht stisfies lier recurrece reltio with polyomil coefficiets. If for give hypergeometric fuctio H,, there exists polyomil P, i of the form P, L L, for some o-egtive iteger L, d P-recursive fuctios 0,..., L such tht F, : H, P, stisfies 4 with some fuctio G, the pir F, G is clled Mrov-WZ pir ssocited with the erel H, MWZ-pir for short. We cll G, MWZ mte of F,. If L 0, the F, G is simply WZ-pir. I 005, M. Mohmmed [5] showed tht for y pure-hypergeometric erel H,, there exists o-egtive iteger L d polyomil P, s bove such tht F, H, P, hs MWZ mte G, F, Q,, where Q, is rtio of two P-recursive fuctios. Pper [6] is ccompied by the Mple pcge MrovWZ which, for give H, outputs the polyomil P, d the G, s bove. From reltio 4 we get the followig summtio formuls. Propositio A. [5, Theorem b] Let F, G be MWZ-pir. If lim F, 0 for every 0, the F 0, lim wheever both sides coverge. Propositio B. [5, Cor. ] Let F, G be MWZ-pir. If lim G, G, 0, 5 G, 0, the F 0, F, + G, +, 6 wheever both sides coverge. Formuls 5, 6 with pproprite choice of MWZ-pirs c be used to covert give hypergeometric series ito differet rpidly covergig oe. As usul, let λ ν be the Pochhmmer symbol or the shifted fctoril defied by λ ν Γλ + ν Γλ {, ν 0; λλ +... λ + ν, ν N. I 979, D. Leshchier [] proved ccelerted series for the vlues β + i the spirit of Apéry s series. Nmely, he showed tht ν A ν ν β + 6 f + ν+ +, 0,,,... 7 ν0

7 Series ccelertio formuls for bet vlues 9 where f 0, f r 0 if r < 0 or r, d f r r 0<l <...<l r< j A ν l j, r, { 3/4 if ν 0 if ν > 0. Formuls, 7 give the followig Apéry-lie series for odd powers of π : π 3 3 π π m0 l0 Usig the geertig fuctio for odd bet vlues 6 +, 6 + m m + l +. m +, m0 m + β + + / + / /4 8 it is esily see tht formuls 7 re geerted by the followig idetity. Propositio For y complex with < we hve β m0 m +. 9 Proof. Idetity 9 c esily be proved by the Mrov-Wilf-Zeilberger method. Tig the erel H, d pplyig the Mple pcge MrovWZ we get tht F,! + + / +! + + / /4! + + +! + + +

8 30 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood d give WZ-pir, i.e., G,! ! F +, F, G, + G,. Now by Propositio A, we get which implies 9. F 0, G, 0, Lemm For y complex, si π m0 m +. 0 Proof. Applyig the trsformtio see [6,.8] i terms of the Guss hypergeometric fuctio + si z si z F, ; 3 ; si z with z π/6 we get the desired represettio. Remr. Formul 0 c be cosidered s expsio of the fuctio siπ/6 ito Newto s iterpoltio series with iterpoltio poits t zero d odd itegers. Theorem Let be complex umber, distict from odd iteger. The π 4 cosπ/ 3 π si m0 If <, the the left-hd side of hs the followig series expsio: I prticulr, we get E + π π3 + 3! π π π , m0 m +. m +.

9 Series ccelertio formuls for bet vlues 3 Proof. By Propositio d Lemm we get β + 3 π si 6 + O the other hd, by 8, we hve β + g + g + m0 where gz / + z. Usig the reflectio formul see [6,.8] we get gz + g z π siπz β + π 4 cosπ/,, m +. from which formul follows. Expdig the left-hd side of, by, we get the required expsio. Comprig d 9 we c esily see tht these idetities produce differet series for the odd powers of π. The series geerted by re simpler i tht sese they ivolve fewer summtios the the correspodig series geerted by 9. Moreover, the formuls obtied here led to some o-trivil redudcy reltios tht c be writte dow explicitly by comprig 7 d the correspodig series for β +,, give by. As exmple, comprig series for π 3 yields the followig redudcy formul: 6 + m0 m Geertig fuctio idetities for eve bet vlues. I this sectio we derive series ccelertio formule for eve bet vlues by pplictio of the Mrov- WZ method. Theorem Let d d be complex umbers such tht d +, 0,,,..., d d distict from zero d egtive itegers. The + d d + + d + d + d + + d.

10 3 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood Proof. Cosiderig the erel H, d + + d d d we get tht d F, d + + d + d d G, d + d d give WZ-pir, d by Propositio A, we obti 3 + d + + d + d d F 0, G, 0 implyig. I prticulr, from if d we get Leshchier s idetity 0. If d / we get the followig idetity geertig ccelerted series for eve bet vlues. Corollry Let be complex umber with <. The β + 8 I prticulr, if 0 we get G Theorem 3 Let be complex umber with <. The I prticulr, β + 64 G Proof. Strtig with the erel + H, m 4m. m /4m m /m

11 Series ccelertio formuls for bet vlues 33 d pplyig the Mple pcge MrovWZ [6], we see tht F, ! ++ d G, ! is Mrov-WZ pir. By usig Propositio A, we get F 0, G, 0, which implies the required sttemet. The geerlized idetities of Corollry d Theorem 3 geerte fst coverget series for β + coverget t geometric rte with rtio /4. Formul 3 ws erlier foud by Lups i [3]. The ext corollry gives much fster coverget series t the geometric rte with rtio /3 6 4/ Corollry Let be complex umber with <. The β p, m /4m 3 m /m, where p, / /4. I prticulr, G Proof. By pplyig Propositio B to the MWZ pir from the proof of Theorem 3 we get or F 0, F, + G, +, β ! p +,, where p, is defied s bove. After simplifyig d replcig Pochhmmer s symbols by biomil coefficiets we get the desired idetity. The ext theorem gives eve much fster coverget series t the geometric rte with rtio 0.

12 34 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood Theorem 4 Let be complex umber with <. The β + 64 where q, m /4m m /m, q, I prticulr, where G q , q Proof. We first observe tht the geertig fuctio of eve bet vlues c be writte i the form β + + Defie the fuctio 3 H, /4 /6 + 3/4 / The usig the Mple pcge MrovWZ we obti tht d F, 6 +! + H, G,! H, is Mrov-WZ pir correspodig to the erel H,. Now by Propositio B, we hve F, + G, + F 0, H0, 3 β +.

13 Series ccelertio formuls for bet vlues 35 or 3! β q +,, where q, is defied s bove. After simplifyig d replcig Pochhmmer s symbols by biomil coefficiets we get the required idetity. Note tht pplictio of Propositio A to the Mrov-WZ pir foud bove recovers Theorem 3. I coclusio we ote tht the differet pproches described i the itroductio for studyig d fidig vrious Apéry-lie series for zet vlues re pplicble to the Dirichlet bet fuctio s well. Hece, it would be of iterest to fid more geerl idetities for bet vlues by mes of bivrite geertig fuctios s it ws doe i the cse of the zet fuctio see for istce idetities 5, 7. Acowledgemets Both uthors th the Istitut de Hutes Études Scietifiques, Bures-sur-Yvette, Frce, where this reserch ws crried out for hospitlity d excellet worig coditios. Refereces [] G. Almvist, A. Grville, Borwei d Brdley s Apéry-lie formule for ζ4 + 3, Experimet. Mth , [] T. Amdeberh, D. Zeilberger, Hypergeometric series ccelertio vi the WZ method, Electro. J. Combitorics 4 997, #R3. [3] J. M. Borwei, D. M. Brdley, Empiriclly determied Apéry-lie formule for ζ4 + 3, Experimet. Mth , o. 3, [4] J. M. Borwei, D. M. Brdley, Serchig symboliclly for Apéry-lie formule for vlues of the Riem zet fuctio, SIGSAM Bulleti of Algebric d Symbolic Mipultio, Associtio of Computig Mchiery, , o., 7. [5] D. H. Biley, J. M. Borwei, D. M. Brdley, Experimetl determitio of Apéry-lie idetities for ζ +, Experimet. Mth , o. 3, [6] H. Btem d A. Erdélyi, Higher Trscedetl Fuctios, Vol., McGrw-Hill, New Yor, 953. [7] D. M. Brdley, Hypergeometric fuctios relted to series ccelertio formuls, Cotemporry Mth , 3 5. [8] Kh. Hessmi Pilehrood, T. Hessmi Pilehrood, Geertig fuctio idetities for ζ+, ζ+3 vi the WZ-method, Electro. J. Combi , o., Reserch Pper 35, 9 pp.

14 36 Kh. Hessmi Pilehrood d T. Hessmi Pilehrood [9] Kh. Hessmi Pilehrood, T. Hessmi Pilehrood, Simulteous geertio for zet vlues by the Mrov-WZ method, Discrete Mth. Theor. Comput. Sci , o. 3, 5 3. [0] Kh. Hessmi Pilehrood, T. Hessmi Pilehrood, A q-log of the Biley-Borwei-Brdley idetity, rxiv: [mth.nt] [] M. Koecher, Letter Germ, Mth. Itelligecer, 979/980, o., [] D. Leshchier, Some ew idetities for ζ, J. Number Theory, 3 98, [3] A. Lups, Formule for some clssicl costts, I Proceedigs of ROGER plouffe/rticles/lups.pdf [4] A. A. Mroff, Mémoiré sur l trsformtio de séries peu covergetes e séries tres covergetes, Mém. de l Acd. Imp. Sci. de St. Pétersbourg, t. XXXVII, No.9 890, 8pp. Avilble t sergey/reserch/history/mrov/mrov890.html [5] M. Mohmmed, Ifiite fmilies of ccelerted series for some clssicl costts by the Mrov- WZ method, Discrete Mthemtics d Theoreticl Computer Sciece 7 005, 4. [6] M. Mohmmed, D. Zeilberger, The Mrov-WZ method, Electroic J. Combitorics 004, #R53. [7] A. v der Poorte, A proof tht Euler missed... Apéry s proof of the irrtiolity of ζ3. A iforml report, Mth. Itelligecer 978/79, o. 4, [8] T. Rivol, Simulteous geertio of Koecher d Almvist-Grville s Apéry-lie formule, Experimet. Mth., 3 004,