On harmonic binomial series

Transcription

1 O harmoic biomial series arxiv:82.766v [math-ph] 9 Dec 28 Mark W. Coffey Departmet of Physics Colorado School of Mies Golde, CO 84 Received 28 April 29, 28 Abstract We evaluate biomial series with harmoic umber coefficiets, providig recursio relatios, itegral represetatios, ad several examples. The results are of iterest to aalytic umber theory, the aalysis of algorithms, ad calculatios of theoretical physics, as well as other applicatios. Key words ad phrases harmoic umber, biomial coefficiet, digamma fuctio, polygamma fuctio, geeralized hypergeometric fuctio, geeralized harmoic umber, Legedre fuctio 5A, 33C2, 33B5 AMS classificatio umbers

2 Itroductio The evaluatio of harmoic umber sums has bee useful i aalytic umber theory for some time e.g., [2, 3]. Recetly, the evaluatio of Euler sums [6, ] has bee importat i various areas of theoretical physics, icludig i support of Feyma diagram calculatios e.g., [7]. More recetly, the performace of geeralized harmoic umber sums has bee useful i evaluatig Feyma diagram cotributios of perturbative quatum field theory [8]. I additio, harmoic umber sums ofte arise i the aalysis of algorithms. This especially applies to algorithms ivolvig searchig, sortig, or permutatios e.g., [4, 8]. Here we are iterested to evaluate sums cotaiig simultaeously two types of special umbers of eumerative combiatorics: biomial coefficiets ad harmoic umbers. Our methods complemet those of Refs. [5, 5]. The aalytic approach of [5] is based upo hypergeometric summatio ad relies o idetities of geeralized hypergeometric fuctios p+ F p at the very special argumet of. These idetities iclude the Chu-Vadermode formula for 2 F, the Pfaff-Saalschütz theorem for 3 F 2, the Dougall-Dixo theorem for 5 F 4, ad the Whipple trasformatio for 7 F 6. I Ref. [5], symbolic computatio usig the Newto-Adrews-Zeilberger algorithm was used to discover harmoic umber relatios. These relatios were motivated by cosideratios to prove certai supercogrueces for Apéry umbers. Our results ot oly complemet those of Ref. [5, 5], but demostrate that various geeralizatios are possible, icludig the itroductio of a summatio parameter. The latter feature meas that may previous results may be cosidered special cases, 2

3 ad poits out that may more results should be achievable i the future. After first itroducig some otatio, we show how various recursio relatios for the sums of iterest may be developed directly. We the illustrate the developmet of itegral represetatios for these sums, ad provide examples. To emphasize the usefuless of the itegral represetatios, we preset low order cases explicitly. I additio, we give a represetatio of biomial-harmoic umber sums i terms of the geeralized hypergeometric fuctio ad its derivative. I particular, with regard to the Gauss hypergeometric fuctio, we are the able to develop results i terms of Legedre fuctio P ν. I the fial sectio, we itroduce a variatio. Namely, we cosider biomial harmoic sums over the order of the geeralized harmoic umbers. Additioal cosideratios i aalytic umber theory motivate the study of these ad related sums [9]. I particular, a certai combiatio of these sums represets a trucatio of a domiatig sum for the Li/Keiper costats. We put H k= /k for the usual harmoic umbers, ad H r for the geeralized harmoic umbers H r These are give i terms of polygamma fuctios ψ as r, H H.. H r = r [ ψ r + ψ r ],.2 r! where ψ r = r r!ζr ad ζ is the Riema zeta fuctio. The asymptotic form of H is well kow, H = l + γ + o, where γ is the Euler costat. 3

4 Ideed, by Euler-Maclauri summatio we have H = l + γ P x dx,.3 x 2 where the periodic Beroulli polyomial P x B x [x] = x [x] /2. The asymptotic form of H r for large is immediately kow from that of ψ r +, ad we have e.g., [], p. 26 [! ψ z = +! ] z 2z + O..4 + z +2 We ivestigate here a subclass of geeralized harmoic umber sums S p q, r, m, z = r p [H q ] m z, z..5 I particular, i this paper we restrict for the most part to m =. Whe, i additio, q = r =, we write S p z = p H Whe simply p =, we omit the superscript. z, z..6 We ote a very coveiet recursio for the geeral sums of Eq..5: S p+ q, r, m, z = z z Sp q, r, m, z..7 I this way, from a iitial sum, successive sums may be obtaied. Although we do ot follow this lie of iquiry, we metio that may itegral represetatios for biomial coefficiets are kow. These e.g., [], pp ; 4

5 [3], p. 53 iclude [], p. 375 = 2+2 π/2 cos x si x si 2mx dx = 2+2 π/2 cos x cosx cos 2mx dx. m π π.8 As a cotour itegral, we have for complex α ad itegral, α = + z α z dz, < r <..9 2πi z =r We may ote that this equatio is particularly well formed for takig derivatives with respect to the parameter α. If both ad are itegral, we o loger have a brach poit at z =, ad may write = + z z dz, < r <.. 2πi z =r Equatios.9 ad. may be immediately verified by usig the biomial theorem to compute the residue of the itegrad at z =. We have Recursio relatios for biomial-harmoic umber sums Propositio. We have the recursio relatio a S + = 2S ,. ad b S + z = + zs z + z For part a, S =, ad for part b, S z = z. 5

6 Propositio 2. We have the recursio relatio a S + p = 2S p + ad b p l= p S l + l 2 l= p l= S +z p p = +zs p z+z p S l l + 2 z2 p l+f l 2, 2,..., 2, ;,,...,, 3;, l.3 p l= For part a, S p =, ad for part b, S p z = z. Propositio 3. We have the recursio relatio a p l+f l 2, 2,..., 2, ;,,...,, 3; z. l.4 S z = zs z + + z..5 Let β p, z The we have b z p = z p F p 2 2, 2,..., 2, ;,,..., ; z..6 p S p z = z p l l= S l z + β p, z..7 Proof of Propositio. The proof of part b subsumes that of part a. By usig a recursio relatio for the biomial coefficiet [] p. 822, we have S + z = + = [ ] + + H z = H + z = = S z H z,.8

7 where we have used the facts + = = H. Further, by shiftig the summatio idex ad usig the recursio relatio for harmoic umbers, we have S + z = S z + + z =H + = S z + H + z + + = = + zs z + z + +,.9 + wherei the latter sum may be obtaied by itegratig the biomial theorem. Proof of Propositio 2a. For this part, omittig the z = argumet, we have [ S p = p + H = p H + = = + = S p + p H = S p + + = p H + p p = S p + l H + l + = l= ].. We ext iterchage the order of the two sums, ad to complete this part of the Propositio we eed to perform the sum l + = = [ 2 ] l, where a = Γa+/Γa is the Pochhammer symbol, ad Γ is the Gamma fuctio. We further use 2 /3 = 2/ + 2 ad = + + +! + = +!..2 7

8 Appealig to the series defiitio of l+ F l completes part a. The steps for part b are very similar, ad are omitted. Proof of Propositio 3a. We use the property =, so that = 2 S z = + H z H z + H z + = H z + + H z + = H z + H z + H z + z z z [ = zs z + ] + z,.3 wherei the latter sum may be obtaied by itegratig the biomial theorem. For part b, we proceed similarly, z = p H z S p = = = = β p, z + p H z H z + p H z p + H = + p 8 z + p H z H z +..4

9 We the biomially expad, iterchage sums, ad the Propositio is complete. Remark. Similar cosideratios ca be give for quadratic ad more geerally oliear sums, that are outside of the preset ivestigatio. Examples The recursio relatios. ad.2 with iitial coditio may be explicitly solved. For Eq.. we have Sice easily we have we obtai S = 2 H F, ; + 2; 2 l F,, + 2; 2 l 2 = 2 S = 2 H a kow result [5]. For Eq..2 we fid S z = +z [H + 2,.6,.7 2 ] z z F, + ; + 2; + l. z + z +.8 By usig a trasformatio formula [] p. 43, we may write this as S z = + z [H + zz F, ; + 2; ] z + l. z z +.9 For z = here we have the explicit reductio to Eqs..5 ad.7. At z = we have 2F, ; + 2; = Γ + 2Γ Γ 2 + =

10 Therefore, we obtai S = /. From Eqs..5 ad.8 we obtai S z = z + z [ H + ] z z + 2 F, ; + ; + l. z + z +.2 Remark. From the recursio relatios of Propositio 2, it appears that the portios of the respective biomial-harmoic series cotaiig harmoic umbers are give for p by ad S p z = +z p p S p = 2 p p H p p,.22 2 [ H p + zz + p + p 2 F, ; p + 2; ] z + l. z z.23 Itegral represetatios for liear biomial-harmoic umber sums We have Propositio 4. We have the itegral represetatio a ad b S p z = z [t p F p 2, 2,..., 2, ;,,..., ; zt p F p 2, 2,..., 2, ;,,..., ; z] dt t, 2. S p q,,, z = q q! z [t p F p 2, 2,..., 2, ;,,..., ; zt p F p 2, 2,..., 2, ;,,..., ; z] lq t dt. 2.2 t

11 Proof. For part a, we employ the relatio H = ψ + + γ, where ψ = Γ /Γ is the digamma fuctio, together with a itegral represetatio for this fuctio [] p. 943: z = p [ψ + + γ] z = = = p z t dt. 2.3 t S p The itegral is absolutely coverget ad we may iterchage summatio ad itegratio. We the rewrite the summatio, usig relatio.2, ad apply the series defiitio of p F p. Equatio 2. follows. For part b, we use Eq..2, writig S p q q,, z = q! = q q! p [ψ q + ψ q ] z = p z = t l q t dt. 2.4 t Here, we employed the result of multiply differetiatig a itegral represetatio for the digamma fuctio to obtai that for the polygamma fuctio. The itegral is agai absolutely coverget ad we may iterchage it with the summatio. Carryig out the summatio, we have Eq Remarks. The itegral represetatio of the polygamma fuctio used above, ψ q z = t z l q t dt = q q!ζq, z, 2.5 t shows the coectio with the Hurwitz zeta fuctio ζs, a at iteger values of q.

12 Referece [8] cotais a umber of other itegral represetatios for harmoic umbers. Similarly, these may be applied to obtai other itegral represetatios of the sums S p. The direct verificatio of the geeral property.7 from the itegral represetatios requires the result z p F p 2, 2,..., 2, ;,,..., ; zt = 2 p t p F p 3, 3,..., 3, 2 ; 2, 2,..., 2; zt, 2.6 as well as a trasformatio formula for p F p. It is possible that this trasformatio may be effected by otig [ 3 2 ] p = + p = [ ] p 2 +, p = To show the utility of itegral represetatios for biomial-harmoic umber sums, we specialize i the followig two sectios to low order cases of the sums S p, ad provide correspodig details. We have Propositio 5. We have for z, givig Explicit expressios for sums S p z S z = z + z 3F 2,, ; 2, 2; z, 3. + z 2

13 Corollary. ad S z = + z 2 S 2 z = z+z 3 { { + z + z [ + z + 2 z 2 z } 3F 2,, ; 2, 2;, + z 3.2a z ] + 2 z + z 3 F 2,, ; 2, 2; The right side of Eq. 3. is easily verified to be a polyomial of degree i z, as it must. For the termiatig 3 F 2 fuctio is a polyomial of degree i z/ + z. Whe multiplied by the prefactor of z + z, we obtai a polyomial of degree. We could cotiue Corollary idefiitely, but these istaces serve our curret purpose. Proof. As with Eq. 2.3 at p =, we have S z = = Chagig variable with u = t, we have z t t dt = [ + z + zt dt ] t. 3.3 S z = + z ad the obtai the Propositio. Corollary follows by applyig property.7. { [ zu ] } du + z u, 3.4 z }. + z 3.2b 3

14 Explicit expressios for some sums S p i terms of harmoic umbers We have Propositio 6. We have ad S 2 S S = 2 H = 2 H = H 2 2, 2 + 2, 2 + [2 2 2 ] 2. 4.a 4.b 4.c Although Eqs. 4.a ad 4.b are kow results [5], our method of proof is differet. I additio, the latter appears to be a rediscovery of earlier closed form summatios [3]. Proof. Usig the itegral represetatio of the previous sectio, writig Eqs. 3.3 ad 3.4 at z =, we have S = [ 2 + t dt ] t [ = 2 u ] du 2 u = 2 [ψ w ] + + γ + dw l 2 /2 w ] = 2 [H F,, + 2; l Here we have applied Eqs. A5 ad A6 of Ref. [9]. By Eq..6, Eq. 4.a follows. 4

15 Writig Eq. 2.3 at z = ad p =, we have = z t t dt S = Chagig variable with u = t, we have S = 2 givig Eq. 4.b. = [t + t 2 dt ] t [ t + ] dt = 2 t 2 t. 4.3 [ u 2 = 2 H ] du u 2 u 2 du 2 2, Followig a similar procedure, writig Eq. 2.3 at z = ad p = 2, we have = Now with u = t, we have S 2 = = 2 z t t dt [t + t + t dt + ] t [ t + 2 = 2 2 t + t + 2 S 2 = 2 2 = 2 2 { + { u[ + u] [ u 2 2 ] du u + ] dt t. 4.5 u } 2 du + 2 u u } 2 [ u]du 5

16 = H } [ 2 + ]. 4.6 Simplifyig the latter terms o the right side of this equatio gives Eq. 4.c, ad completes the Propositio. We have Propositio 7. For itegers p, Hypergeometric approach S, p,, = x x p ν p F p [ ν,..., ν;,...,; p x] ν= +H x pf p [,..., ;,..., ; p x]. 5. Proof. We have ν = ν, 5.2 so that Usig = p ν x = p F p [,..., ;,...,; p x]. 5.3 ν νp = p ν p [ψν + ψν + ], 5.4 ad the sum = p H x = x m= we obtai the Propositio. p H m x m = x S, p,,, 5.5 m x 6

17 We focus o the p = 2 case of Propositio 7, ad obtai a umber of series represetatios. For this, we itroduce the Legedre fuctios of the first kid P ν, ad the fuctio R z = P νz ν + z l ν= 2 P z, 5.6 where is a iteger, ad P is a Legedre polyomial. We have Propositio 8. S, 2,, = x x 2 ν 2 F ν, ν; ; x ν= = x 2 x R +H x 2F, ; ; x. + x x + x + H x x P. 5.7 x Proof. We first observe that through the use of a trasformatio formula for 2 F we have P ν 2x = 2 F ν, ν + ; ; x = x ν 2F ν, ν; ; x, 5.8 x givig z + P ν z = 2 ν 2F ν, ν; ; z. 5.9 z + Differetiatio of this equatio with respect to ν ad compariso with the defiig relatio 5.6 yields z + R z = 2 ν 2 F With a chage of variable, we obtai Eq ν, ν; ; z z + ν=. 5. 7

18 We ow discuss ad illustrate Propositio 8, i the course of which we obtai Corollary 2 S, 2,, = recoverig a kow result [5]. =H 2 = 2H H 2 2, 5. There are several series represetatios of the fuctio R available, icludig the Bromwich forms [4] R z = k= k [P kz P k z]p k z, 5.2 R z = 2 +k 2k + k + k + [P kz P z], 5.3 k= ad from a formula of Jolliffe [2], + z R z = 2 l P z + 2 2! d dz [ z z + l z + 2 ]. 5.4 Besides these older results, R ad closely related polyomials have bee very recetly restudied [7]. Sice the Legedre polyomial has P = ad P k = k, we readily see that the fuctio R satisfies R = ad R = 2 H. It is evidet from the Bromwich formula 5.3 that R z = 2H 2 H P z + 2 +k 2k + k + k + P kz. 5.5 k= The property R = 2 H follows immediately from this equatio. With the use of the duplicatio formula satisfied by the digamma fuctio, the property R = may also be deduced from Eq

19 Equivalet to the Chu-Vadermode summatio 2 F, ; ; = 2, is the relatio, via Eq. 5.9, lim x x P + x = x I additio, with the help of Eq. 5.5 ad the use of partial fractios, we have + x 2 lim x x R = 2 H 2 H. 5.7 x Therefore, from Propositio 8 we obtai Corollary 2. By the same method of this sectio, o takig more derivatives with respect to ν i Eq. 5., it is possible to get higher values of q i the sums defied i Eq..5. I this regard, we very briefly metio the fuctios Qx, z = 2 F x, x; ; z = ad the expasio = x x! 2 z, Qx Qx,, 5.8 Qx = + A 2 x 2, 5.9 writte i Ref. [6]. Obviously the Maclauri coefficiets here are give by A 2 = 2 d Qx ! dx x= The by the property 5.4 at p =, the coefficiets A 2 ivolve sums ad products of geeralized harmoic umbers. Further coectios with Legedre fuctios ad Stirlig umbers of the first kid we do ot pursue here. Amog the geeralized hypergeometric fuctios that reduce to polyomials is the family H x, a, z = 3 F 2, +, x;, a; z, 5.2 9

20 where is a iteger ad x, a C, with a, 2,... These fuctios may be obtaied by a certai itegratio over shifted Legedre polyomials, ad similarly for higher order fuctios p F p. Obviously the set of fuctios of Eq. 5.2 icludes H +,, z ad related fuctios. Oe may the ask for a geeralizatio of Propositio 8 for the case of p = 3 i Propositio 7, a effort that we are leavig to future work. We also metio a itegral represetatio for the sum of Propositio 8, ad how the case of Corollary 2 may be otherwise obtaied. We have S dt, 2,, z = [ 2 F,,, zt 2 F,,, z] t As special case, we have S, 2,, = F,,, t dt t 5.23a 2 = 2 x P 2x dx x 2 = z P z 5.23b dz + z, 5.23c where we chaged variable ad used Eqs. 5.8 ad 5.9. Oe way to demostrate equality with the result of Corollary 2 is to show that the itegral of this equatio satisfies the same recursio relatio Q + Q = /2 + + / / + as the quatity Q = H 2 2H, with iitial coditio Q = /2. We also ote the itegral represetatio Q = t 2 dt t 2

21 Other class of biomial harmoic sums Fially, we describe aother class of sums, where the summatio is ow over the order of the geeralized harmoic umbers, We have S M, z m=2 H m M m zm. 6. Propositio 9. Let L α be the associated Laguerre polyomial of degree e.g., [, 9, ]. We have S M, z = z [ L z lt ] t M dt. 6.2 t Proof. We use relatio.2 together with a itegral represetatio for the polygamma fuctio, S M, z = = m=2 m=2 m [ z m m ψ m M + ψ m ] m m! z m m m! t M l m t dt. 6.3 t Iterchagig the fiite summatio with the itegratio ad applyig the power series defiitio of L, we obtai the Propositio. Remarks. I the limit as M i Eq. 6.2, we have the represetatio lim S M, z = z M This is the same limit i which H m M [ L z l t ] dt t. 6.4 ζm. I particular, we put S = lim M [S M, S M, /2]

22 With a chage of variable, we obtai agreemet with the itegral represetatio of S of Ref. [9] p. 26. I particular, this alteratig biomial sum provides the apparetly domiat cotributio to the Li/Keiper costats λ of the Li criterio for the Riema hypothesis [9]. This sum has bee show to be O l, describig the sigificat cacellatio withi the summad. Propositio. We have M S M, z = + z + + z zh M M. 6.6 =2 Proof. By usig the recursio relatio of geeralized harmoic umbers, we have from the defiitio 6. S M, z = S M, z M z + + z M. 6.7 This recursio may also be foud from the itegral represetatio of Propositio 9. We form the differece [ S M, z S M, z = z L z lt ] t M dt. 6.8 We the itegrate by parts, S M, z S M, z = = M d dt L z l t t M dt z M L z l tt M dt L z M = + z z. 6.9 M M 22

23 I the last step, we may put t = exp v, ad use a extesio of the Laplace trasform of the Laguerre polyomial e.g., formula of [], with λ = z ad µ =. Sice we have the iitial value S, z = z + + z, solutio of the recursio 6.7 gives the Propositio. A alterative proof of Propositio is ow obvious. For from biomial expasio we obtai M + z M = l= = l=2 l z l l = l l= z l H l M l z l H l M + M + zh M. 6. It is possible to expad the factor t of Eqs. 6.2 ad 6.4 as a geometric series, thereby givig aother series represetatio of the sum S, ad providig a third proof of Propositio 9. We have Corollary 3. We have [ S M, z = zh M M + + z + z ]. 6. M + A simple shift of summatio idex here recovers the relatio 6.6. I order to obtai the result 6., we write S M, z = z [L zv ][e M++v e +v ]dv. 6.2 = We the use for Re k >, L zve kv dv = z 23 [ + z ]. 6.3 k

24 H m M Remarks. Corollary 3 must be equivalet to usig the partial fractios form of comig from that for the polygamma fuctio, i 6., ad the iterchagig summatios. By maipulatig S M, z we may obtai related sums. For istace, by itegratig Eq. 6., usig Propositio 8, ad iterchagig itegratios, we have m=2 m H M m m + um = u = u =2 + From Propositio we obtai the combiatio [ L 2 u lt ] t M dt + 2 t + u + M u [ M S M, S M, /2 = ] H M = M [ 2 + ] The latter expressio is the M < form of Eq. 9 of Ref. [9]. Plaily, the sums of this sectio may be geeralized to those such as r S M, p, q, r, z m p [H m M ] q z m. 6.6 m=2 m Ackowledgemet This work was supported i part by Air Force cotract umber FA

25 Refereces [] M. Abramowitz ad I. A. Stegu, Hadbook of Mathematical Fuctios, Natioal Bureau of Stadards 972. [2] B. C. Berdt, Ramaua s otebooks, part I, Spriger 985. [3] B. C. Berdt, Ramaua s otebooks, part IV, Spriger 994. [4] T. J. I A. Bromwich, Certai potetial fuctios ad a ew solutio of Laplace s equatio, Proc. Lodo Math. Soc. 2, [5] W. Chu ad L. De Doo, Hypergeometric series ad harmoic umber idetities, Adv. Appl. Math. 34, [6] M. W. Coffey, O some log-cosie itegrals related to ζ3, ζ4, ad ζ6, J. Comp. Appl. Math. 59, [7] M. W. Coffey, O oe-dimesioal digamma ad polygamma series related to the evaluatio of Feyma diagrams, J. Comp. Appl. Math. 83, [8] M. W. Coffey, O a three-dimesioal symmetric Isig tetrahedro, ad cotributios to the theory of the dilogarithm ad Clause fuctios, J. Math. Phys. 49, , arxiv/math-ph/8273v2. [9] M. W. Coffey, Toward verificatio of the Riema hypothesis: Applicatio of the Li criterio, Math. Physics, Aalysis ad Geometry 8, , arxiv/math-ph/

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