Harmonic Number Identities Via Euler s Transform

Transcription

1 Joural of Iteger Sequeces, Vol ), Article Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio USA -boyadzhiev@ou.edu Abstract We evaluate several biomial trasforms by usig Euler s trasform for power series. I this way we obtai various biomial idetities ivolvig power sums with harmoic umbers. 1 Itroductio ad prerequisites Give a sequece {a }, its biomial trasform {b } is the sequece defied by b 0 or, i the symmetric versio b 0 a, with iversio a 0 1) +1 a with iversio a 1) b, 0 1) +1 b see [7, 12, 14]). The biomial trasform is related to the Euler trasform of series defied i the followig lemma. Euler s trasform is used sometimes for improvig the covergece of certai series [1, 8, 12, 13]. Lemma 1. Give a fuctio aalytical o the uit dis ft) a t, 1) 1

2 the the followig represetatio is true 1 t f Proof ca be foud i the Appedix.) If we have a coverget series t 0 )a. 2) we ca defie the fuctio s a, 3) The, with t 1 2 i 2) we obtai ft) s a t, t < 1. 4) 0 )a ) This formula is a classical versio of Euler s series trasformatio. Sometimes the ew series coverges faster, sometimes ot see the examples i [10]. We shall use Euler s trasform for the evaluatio of several iterestig biomial trasformatios, thus obtaiig biomial idetities of combiatorial ad aalytical character. Evaluatig a biomial trasform is reduced to fidig the Taylor coefficiets of the fuctio o the left had side of 2). I Sectio 2 we obtai several idetities with harmoic umbers. I Sectio 3 we prove Dilcher s formula via Euler s trasform. This paper is close i spirit to the classical article [7] of Hery Gould. Remar 2. The represetatio 2) ca be put i a more flexible equivalet form 1 µt 1 t f t )µ a, 6) 1 t 0 where,µ are appropriate parameters. To show the equivalece of 2) ad 6) we first write f µt ad the apply 2) to the fuctio gt) f µ t). This provides ) ) µ 1 µ f Replacig here t by t yields 6). t µ a t ), 7) t 0 2 ) a ). 8)

3 With 1 ad µ 1 we have 1 t t 1 f t 1 t 0 correspodig to the symmetrical biomial trasform. Lemma 3. Give a formal power series we have gt) gt) ) 1) +1 a, 9) b t, 10) b )t. 11) This is a well-ow property. To prove it we just eed to multiply both sides of 11) by ad simplify the right had side. 2 Idetities with harmoic umbers 0 Propositio 4. The followig expasio holds i a eighborhood of zero log1 αt) 1 βt 1 where α,β are appropriate parameters. αβ α2 β α ) t 12) Proof. It is sufficiet to prove 12) whe β 1 ad the rescale the variable t, i.e. we oly eed log1 αt) α + 12 α ) α t. 13) 1 This follows immediately from Lemma 3. Corollary 5. With α 1 i 13) we obtai the geeratig fuctio of the harmoic umbers log) H t, H ) The ext propositio is oe of our mai results Propositio 6. For every positive iteger ad every two complex umbers,µ, 1 ) H µ H + µ) + µ) µ) ) 3

4 Proof. We apply 6) to the fuctio O the left had side we obtai 1 log1 µt ) 1 t 1 t 1 µt 1 t log) ft) H t. 16) which equals, accordig to Corollary 5 ad Propositio 4, H + µ) t 1 1 At the same time, by Euler s trasform the right had side is log1 + µ)t) log1 t) µ)t 1 + µ)t, 17) ) + µ) µ) t. 18) t 1 1 )H µ. 19) Comparig coefficiets i 18) ad 19) we obtai the desired result. Corollary 7. Settig µ 1 i 15) yields the well-ow idetity see, for istace, [6, 14]): )H 2 1 H. 20) 2 1 Corollary 8. Settig 1 i 15) reduces it to 1 H µ H 1 + µ) 1 + µ) µ) µ ). 21) We shall use this last idetity to obtai a represetatio for the combiatorial sum 1 1 H m µ, 22) by applyig the operator µ d dµ )m to both sides i 21). First, however, we eed the followig lemma. Lemma 9. For every positive iteger m defie the quatities am,,µ) µ d ) m 1 + µ) dµ 0 m µ. 23) 4

5 The am,,µ) 0!Sm,)µ 1 + µ). 24) This is a ow idetity that ca be foud, for example, i [6]. From Lemma 9 we obtai aother of our mai results. Propositio 10. For every two positive itegers m ad, 1 1 H m µ 1 am,,µ)h am,p,µ). 25) p p1 Proof. Apply µ d dµ )m to both sides of 21) ad ote that µ d dµ )m µ m µ. The sums 22) were recetly studied by M. Coffey [3] by usig a differet method a recursive formula) ad a represetatio was give i terms of the hypergeometric fuctio 3 Stirlig fuctios of a egative argumet. Dilcher s formula Some time ago Karl Dilcher obtaied the ice idetity 1 1) 1 1, 1 j m 1 j 2 j m, 26) j 1 j 2 j m as a corollary from a certai multiple series represetatio [4, Corollary 3]; see also a similar result i [5]. As this is oe biomial trasform, it is good to have a direct proof by Euler s trasform method. Before givig such a proof, however, we wat to poit out oe iterestig iterpretatio of the sum o the left had side i 26). Let Sm,) be the Stirlig umbers of the secod id [9]. Butzer et al. [2] defied a extesio Sα,) for ay complex umber α 0. The fuctios Sα,) of the complex variable α are called Stirlig fuctios of the secod id. The extesio is give by the formula Sα,) 1! with Sα, 0) 0. Thus, for m, 1, 1 1) α, 27) 1) 1!S m,) 1 1) 1. 28) m 5

6 For the ext propositio we shall eed the polylogarithmic fuctio [11] Li m t) 1 Propositio 11. For ay iteger m 1 we have t m. 29) 1) 1!S m,) 1 j 1 j 2 j m, 1 j 1 j 2 j m. 30) Proof. The proof is based o the represetatio t Li m t jm, j 1 j 2 j m 1 j 1 j 2 j m, 31) see [15]) from which, i view of Lemma 2, with coefficiets 1 t Li m A t, 32) A 1, j 1 j 2 j m 1 j 1 j 2 j m. 33) The assertio ow follows from 9). I coclusio, may thas to the referee for a correctio ad for some iterestig commets. 4 Appedix We prove Euler s trasform represetatio 2) by usig Cauchy s itegral formula, both for the Taylor coefficiets of a holomorphic fuctio ad for the fuctio itself. Thus, give a holomorphic fuctio f as i 1), we have a 1 2πi L 1 1 f) d, 34) for a appropriate closed curve L aroud the origi. Multiplyig both sides by ) ad summig for we fid 0 a 1 2πi L 0 ) 1 f) d ) f) d. 35) 2πi L Multiplyig this by t with t small eough) ad summig for we arrive at the desired represetatio 2), because t ) ) 1 6 t 1 t, 36)

7 ad therefore, t 0 Refereces )a 1 1 2πi L f) t 1 t d 1 t f. 37) [1] P. Amore, Covergece acceleratio of series through a variatioal approach, J. Math. Aal. Appl ) 2006), [2] P. L. Butzer, A. A. Kilbas ad J. J. Trujillo, Stirlig fuctios of the secod id i the settig of differece ad fractioal calculus, Numerical Fuctioal Aalysis ad Optimizatio ) 2003), [3] Mar W. Coffey, O harmoic biomial series, preprit, December 2008, available at [4] K. Dilcher, Some q series idetities related to divisor factors, Discrete Math ), [5] P. Flajolet ad R. Sedgewic, Melli trasforms ad asymptotics, fiite differeces ad Rice s itegrals, Theor. Comput. Sci ), [6] H. W. Gould, Combiatorial Idetities, Published by the author, Revised editio, [7] H. W. Gould, Series trasformatios for fidig recurreces for sequeces, Fiboacci Q ), [8] J. Guillera ad J. Sodow, Double itegrals ad ifiite products for some classical costats via aalytic cotiuatios of Lerch s trascedet, Ramauja J ), [9] R. L. Graham, D. E. Kuth, ad O. Patashi, Cocrete Mathematics, Addiso-Wesley, New Yor, [10] K. Kopp, Theory ad Applicatio of Ifiite Series, Dover, New Yor, [11] L. Lewi, Polylogarithms ad Associated Fuctios, North-Hollad, Amsterdam, 1981, [12] H. Prodiger, Some iformatio about the biomial trasform, Fiboacci Q ), [13] J. Sodow, Aalytic cotiuatio of Riema s zeta fuctio ad values at egative itegers via Euler s trasformatio of series, Proc. Amer. Math. Soc ), [14] M. Z. Spivey, Combiatorial sums ad fiite differeces, Discrete Math ),

8 [15] E. A. Ulasii, Idetities for geeralized polylogarithms, Mat. Zameti 73 4) 2003), Mathematics Subject Classificatio: Primary 05A20; Secodary 11B73. Keywords: Harmoic umbers, biomial trasform, Euler trasform, Stirlig umber of the secod id, Stirlig fuctio, combiatorial idetity, polylogarithm, Dilcher s formula. Received Jue ; revised versio received July Published i Joural of Iteger Sequeces, August Retur to Joural of Iteger Sequeces home page. 8