Victor Adamchik. Wolfram Research Inc., 100 Trade Center Dr., October 21, 1996

Transcription

1 O Strlg umbers ad Euler sums Vctor Adamch Wolfram Research Ic., Trade Ceter Dr., Chamag, IL 68, USA October, 996 Abstract. I ths aer, we roose the aother yet geeralzato of Strlg umbers of the rst d for o-teger values of ther argumets. We dscuss the aalytc reresetatos of Strlg umbers through harmoc umbers, the geeralzed hyergeometrc fucto ad the logarthmc beta tegral. We reset the te seres volvg Strlg umbers ad demostrate how they are related to Euler sums. Fally we derve the closed form for the multle zeta fucto (; ;:::; ) for >. Itroducto ad otatos. Throughout ths artcle we wll use the followg detos ad otatos. Strlg umbers of the rst d are deed by the recurrece relato (see []) h h, (, ) () wth tal values h + h,, whch related to other otatos for Strlg umbers by h (,), S () (,), s(; ) The Pochhammer symbol s (z) Y (z +, ),(z + ),(z) The geeralzed hyergeometrc fucto s deed by a ;a F ; :::; a q b ;b ;:::;b q ; z Q qq m m jh j (a m ) z (b m )! z j

2 Followg [] we dee "r-order" harmoc umbers by or, terms of olygamma fuctos, H (r) (,)r, (r, )! H (r) Harmoc umbers. r () (r,) ( +), (r,) () We wll cosder the fuctoal equato () ad derve ts soluto terms of harmoc umbers. It s ow ([]) that h (, )! h (, )! () H, h (, )! ((H () 3!,), H,) () h To d we set 4 () ad usg the above tal codtos we obta 4 h 4 (, )! ((H () 3!,) 3, 3H,H () (), +H,) (3) It follows, the, that the geeral formula for Strlg umbers of the rst d terms of harmoc umbers s where the w-sequece s deed recursvely by w(; ) w(; m) h (, )! w(; m, ) (3) m (m, )! m, (, m) H (+), w(; m,, ) It s terestg to observe that for a gve m the umber of summato terms the w-sequece s exactly a umber of arttos of m. The w-sequece ca be rewrtte also through a multle sum w(; m),, + :::, m m,+ m! ::: m Moreover, there s aother reresetato for the w sequece. Cosder the Pochhammer fucto (x,) ad d the value of ts m-th dervatve wth resect

3 to x at the ot x,. Sce the Pochhammer fucto s a rato of Gamma fuctos, oe ca easly rove that lm! x, d m dx m (x, ) w(x; m),(x) (4) O the other had, tag to accout that the Pochhammer symbol s the geeratg fucto for Strlg umbers, we have d m x x lm! x, dx m (x, ) mh (, m)m (,) x, x,,m Equatg the rght sdes of (4) ad ths detty we arrve at w(; m) (, )! m+h (, m)m (,),,m, (5) Comarg (3) ad (5) we derve the followg Strlg umber detty h m m+h,,m (,), m, where + m s a odd teger. At the ed of ths secto we establsh the l betwee the w-sequece ad Strlg olyomals. The latter are deed (see, for examle, [] ) by h m (m,, )! (m) m, m! Comarg ths deto wth formula (3), we get (m) w(m; m,, ) m for ostve tegers ad m. I artcular, for m, ths formula reduces to m m,(m) H () m, 3 Hyergeometrc fuctos. Cosder the hyergeometrc fucto b; a; : : : ; a +F a + ;:::;a+ ; (6) where s a ostve teger ad <(b) <.We shall derve the closed form of ts reresetato ad show that ths d of hyergeometrc fucto ca be regarded as a geeralzato of Strlg umbers of the rst d. For ths urose we eed the Mell-Bares cotour tegral that rovdes us wth the followg reresetato for fucto (6) 3

4 (a) I,(s),(b, s),(a, s) (,),s ds (7),(b) L,(a +, s) where the cotour of the tegrato L s a left loo, begg ad edg at, ad ecrclg all oles of,(s) the ostve drecto, but oe of the oles of,(b, s),(a, s). Alyg,(b, s) (,),s,(, b),(b),(, b + s) + e,b,(b, s),(s),(, s) to the tegrad (7) we obta I L I L,(s),(b, s),(a, s),(a +, s) (,),s ds,(s),(a, s),(, b + s),(a +, s) ds +,(b, s),(a, s) IL e, b,(, s),(a +, s) ds Observe that the secod tegral the rght sde s zero sce ts tegrad has o sgulartes sde the cotour L. For the rst tegral we shall use the rcle of the aalytc cotuato of Mell-Bares tegrals whch states that I L,(s),(a, s),(, b + s),(a +, s) ds, IM,(s),(a, s),(, b + s),(a +, s) ds where M s a rght loo, begg ad edg at + ad ecrclg all oles of,(a,s) the egatve drecto, but oe of the oles of,(s). By usg the resdue theorem we obta Proosto If IN, b 6 IN ad <(b) <the +F b; a; : : : ; a a + ;:::;a+ ; (,) + (a),(, b) (, )! d,,,( + s)(a, s) lm s! a ds,,(, b + + s)(a, s, ) (8) Cosder two secal cases of ths formula. Let,b a, a<ad a s ot a teger. The formula (8) becomes +F a; a; :::; a a +;:::;a+; (,), a w(a;, ) s(a)(, )! (9) 4

5 If a s a ostve teger the from (9) t follows that +F a; a; :::; a a +;:::;a+; a h a, a (,) a,, (, ) (a, )! + Notce that formulas (8), (9) ad () geeralze (7...6)-(7...7) from [3]. Now we are ready to dee Strlg umbers of the rst d for o-teger values of the uer argumet. Combg formulas (3) ad (9) we arrve at Proosto If <(z) <, ad IN the h z (,) + F z,(, z) + or the seres form h z (,) + s(z) z; z; :::; z z +;:::;z+; () (),( + z)!( + z) () The followg secal values ca be easly comuted: h, log(4) h 3 6 ( + 3 log (4)) h, 4 6 ( log(4) + log 3 (4) + (3)) Sce the secod argumet of Strlg umbers s actually a order of the hyergeometrc fucto, t should always be a ostve teger. I the ext secto we d such ategral reresetato for Strlg umbers that allows us to geeralze Strlg umbers wth resect to both argumets. 4 Itegral reresetatos. Our rcal am s to derve the aalytc cotuato of Strlg umbers for comlex values of ther h arameters. For ths urose we cosder rst the very artcular case, amely.from the secto follows that h (, )! ( + ()) (3) We shall mae of use the well-ow tegral reresetato of olygamma fuctos: () (z), () () (,) + Z t z,,, t log ( ) dt (4) t 5

6 Substtutg (4) to (3), we d h z,(z) Z, t z,, t dt; <(z) > (5) I order to d the tegral reresetato of Strlg umbers of hgh orders we rewrte (4) as Z () t z, (z) (,) +, t log ( t ) dt assumg that <() >. We observe that the tegrad ca be obtaed by deretato wth resect to the dummy varable s t z, log ( (, t) r t ) lm d s! ds (( t z, t )s ) (, t) r We added the ew arameter r here to rovde the covergece of the tegral at uty. Chagg the order of tegrato ad deretato we have Z d t z,s, dt d,(, r),(z, s) ds (, t) s ds,( + z, s, r) If we set the varable arameter r to z, the rght sde wll closely resemble equato (4) wth z, ad x z, s. Therefore, by equato (9) t follows that +F z; z; :::; z z +;:::;z+; z (, )! Z t z, (, t) z log, ( t ) dt However, vew of formula (), the above tegral gves us aother aroach to the geeralzato of Strlg umbers for comlex values of ther argumets. Proosto 3 If < <(z) < <(), the h z,(, z),() Z t z, (, t) z log, (t) dt (6) I [4] ad [5] the smlar tegral reresetato has bee obtaed for the Strlg umbers s(z;). Though these two forms of Strlg umbers are related by s(z;) (,) z,h z where z ad are teger, however, for o-teger z ad ther tegral geeralzatos are qute deret. Now we shall verfy f ths h tegral reresetato (6) satses the fuctoal, equato (). Begg wth ad alyg a tegrato by arts we get, 6

7 h,,,(, ),(, ),(),(, ) Z Z t, (, t), log, (t) dt t, (, t), d(log, (t)) Z,(),(, ) ((, ) t, (, t), log, (t) dt + Z t, (, ) (, t) log, (t) dt) h, h,(, ) + We omtted the o-tegral term whch salways zero assumg <() > <() >. The tegral reresetato (6) ca be aalytcally cotued ( the Hadamard sese) to the half-lae <(z) <. For smlcty we show here how to mae the cotuato to the str, < <(z). For such z, the tegrad (6) has a o-tegrable sgularty at the ot t whch ca be removed by substractg from (, t),z the rst term of ts Taylor`s exaso. Cosequetly, formula (6) yelds h z,(, z),() Z t z, ((, t),z, ) log, (t) dt + Z,(, z),() :f: t z, log, (t) dt It s easy to see that the tegrad the secod tegral ossess the Hadamard roerty at the ot t : :f: Z Fally we have h z,(, z),() where, < <(z) < <(). t z, log, (t) dt (,),,() z Z t z, ((, t),z, ) log, (t) dt + (,), z,(, z) (7) 5 Seres volvg Strlg umbers. I ths secto we cosder the followg tye of sums volvg Strlg umbers h G ;q (8)! q 7

8 Let us beg wth the smle examle h! Usg the tegral reresetato (5) ad chagg the order of summato ad tegrato, we get Z! t, 6L (t) dt (3) 6 t (, t) h From ths detty oe would exect the atter to rema uchaged ad so that: G ; h! ( +) (9) To rove ths formula we shall use the method of geeratg fuctos. We have to show that t ( +) t h! The left sde of ths equalty s a well-ow sum (see [6] (54.3.)): t ( +),, (, t) I the rght sde, chagg the order of summato ad tag to accout the formula (5..) from [6], we have t h!! t h (t)!,, (, t) I a smlar way, usg the tegral reresetato (5), we obta Proosto 4 If IN, the G ;q h (q +) q! (q +), q q, ( +)(q +, ) () Ite seres of ths d ca be vewed as the artcular cases of the Nelse geeralzed olylogarthm S ; (z): S ; (z) (, )!! Z (, log(t)), (, log(, zt)) dt t h extesvely studed [7]. It was show there, for examle, that z! () z h! ( +)+ (,), L! +,(, z) log (, z) () 8

9 From the olylogarthmc tegral (), erformg oe tme tegrato by arts, t s easy to derve the symmetry roerty of Strlg sums: G ;q G q; (3) Based o the tegral reresetato () may terestg sums volvg Strlg umbers of the rst d ca be derved. Here are some of them: h (,) +;, log()), +( ( + )!! h ( + )! h ! 5, (3) 3 h 3!, 4 (3), (5) + 5 (7) h ! 68 + (3), 3 (3) (5) 3 I geeral, Strlg sums G ;q ca always be rereseted te terms of Zeta fuctos: G ;q (,)q, (q, )!! lm! lm! d, q +,(, ),( + ) d d q,,(, + ) The formula follows straghtforwardly from the tegral reresetato (). (4) 6 Euler sums. I ths secto we establsh a coecto betwe Strlg sums G ;q ad Euler sums. The Euler sum of the weght e + e + ::: + e + q s deed (see [9]) by S e ;e ; :::;e;q H (e ) H (e ) :::H (e) (5) q h We cosder G ;, ad relace the Strlg umbers by ts reresetato through harmoc umbers from the secto. We obta G ;, h (, )! H (), H (), + S ;, G ; whch ca be rewrtte as 9

10 S ;!(G ;, +G ; ) (6) I the same maer, cosderg G ;,,G 4;, ad G 5;, we arrve the very smle way (comare t wth [8]) at the followg dettes: S ;;, S ;!(G 3;, +G ; ) (7) S ;;;, 3S ;; +S 3; 3!(G 4;, +G 3; ) (8) S ;;;;, 6S ;;; +3S ;; +8S ;3;, 6S 4; 4!(G 5;, +G 4; ) (9) The atter s qute obvous. The coecets by S the left sdes are detcal to those whch are by harmoc umbers reresetatos of Strlg umbers from the secto. Now let us cosder the multle Euler zeta sum (also called Euler/Zager sums) (s ;s ;:::;s d ) > >:::> d s s d ::: s d tesvely studed recet tmes ([], [], [], [3], [4]) ad establsh the closed form reresetato for (; ; ;:::; ), where >. It was ow to Euler that (; ;:::;)G +;, (3) However, t s bee oly few years whe the closed form for (; ; ) was dscovered (see [], [8]). Here let us demostrate aother (ad very smle) aroach to evaluatg of (; ; ). We have (; ; ) Evaluatg the er sum, we obta or sce (; ; ),, 3, H (), (; ; ) ( + ), S ; +, H (), 3,, ( + ), S ; H () (3) where S s deed by (5). To our good fortue, the er sum the rght sde of the detty (3) s summable terms of harmoc umbers.

11 We eed the followg lemma whch ca be roved by usg the tegral reresetato (4) Lemma H () () ((H ) + H () ) (3) Thus, the detty (35) ca be rewrtte (; ; ) ( + ), S ;, S ;+ + S ;; + S ; ad therefore, tag to accout formulas (), (6) ad (7), we have (; ; ) G 3;, (33) where G s deed by (4). As we ow, for o-teger values of the Strlg sum G s deed by the Nelse geeralzed olylogarthm (). Sce that, the above detty ca be rewrtte as (; ; ) S 3;,() (34) whch geeralzes Marett`s formula for o-tegers. Now let us aroach (; ; ; ). Proceedg the smlar way, we obta (; ; ; ) S 3;, ( +3),, H () +, (H () ) +, H () (35) It s uow to me whether or ot the above er sums are doable te terms. Fortuately, they are doable ars. We eed the followg lemmas: Lemma H () + H () H (3) + H () H () (36) Ths lemma s almost obvous. To rove tyou eed to cosder ay oe of the above sums, relace harmoc umbers by the te sum () ad chage the order of summato. Lemma 3 (H () ) + H () () ((H ) 3 +3H () H () +H (3) ) (37) 3 The lemma ca be easly roved by ducto. Now, substtutg (36) ad (37) to (35) ad tag to accout formulas (6),(7) ad (8), we mmedately arrve at

12 Proosto 5 (; ; ; ) G 4;, S 4;,() (38) Comarg formulas (3), (33) ad (38), we ca derve the geeral reresetato for (; ;:::;) Proosto 6 (; ;:::;)G +;, S +;,() (39) where s a umber of 's argumets of the multle zeta fucto. Acowledgemet. I would le to tha R. Cradall ad B. Goser for helful dscussos. Refereces [] R. L. Graham, D. E. Kuth, O. Patash (989), Cocrete Mathematcs, Addso-Wesley. [] L. Comtet (974),Advaced Combatorcs,Rechel, Dordrecht ad Bosto. [3] A. P. Prudov, Yu. A. Brychov, O. I. Marchev (99), Itegrals ad Seres, Vol. 3: More Secal Fuctos, Gordo ad Breach, New Yor. [4] P.L.Butzer, M.Hauss (99), Strlg umbers of rst ad secod d, I "Aroxmato, Iterolato ad Summablty", Israel Math Cof. Proceedgs, Wezma Press, Israel, Vol. 4, 99, [5] P.L.Butzer, M.Hauss (99), Rema zeat fucto, Al. Math. Lett. 5(), 99, [6] E. R. Hase (975),A Table of Seres ad Products, Pretce-Hall, Ic., N.J. [7] K.S.Kolbg (986),Nelse`s geeralzed olylogarthms, SIAM J. Math. Aal., 7 (5), [8] D. Borwe, J. Borwe, R. Grgesoh (995),Exlct evaluato of Euler sums, Proceedgs of the Edburgh Mathematcal Socety 38, [9] F. Flajolet, B. Salvy (996), Euler Sums ad Cotour Itegral Reresetatos, Exermetal Mathematcs (to aear). [] D. Zager (994), Values of zeta fuctos ad ther alcatos, rert, Max- Plac. [] C. Marett (994), Trle Sums ad the Rema Zeta Fucto, J. Number Theory, 48,. 3-3.

13 [] R.E. Cradall, J. P. Buhler (994), O the Evaluato of Euler Sums, Exermetal Mathematcs 3, [3] R.E. Cradall (996), Fast evaluato of multle zeta sums, Exermetal Mathematcs (to aear). [4] J. M. Borwe, D. M. Bradley, D. J. Broadhurst (996), Evaluatos of -fold Euler/Zager sums: a comedum of results for arbtrary, mauscrt. 3