ODD MAGNE OGREID AND PER OSLAND

Transcription

1 DESY hep-th/9868 December 997 SUMMING ONE- AND TWO-DIMENSIONAL SERIES RELATED TO THE EULER SERIES arxiv:hep-th/9868v 6 Jan 998 ODD MAGNE OGREID AND PER OSLAND Abstract We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series These include both one-dimensional and two-dimensional series Most of these series can be expressed in terms of ζ, ζ, the Catalan constant G and Cl π/ where Cl θ is Clausen s function Introduction When calculating radiative corrections in Quantum Field Theory, one will encounter multi-dimensional Feynman integrals [] These often present considerable mathematical challenges Several methods are available for such calculations Unfortunately, the evaluation of such integrals is a tedious tas and a simple result cannot always be found A powerful method for doing Feynman integrals consists in using Mellin transforms [6, 8] This approach is particularly useful when one wants to expand the result in powers of logarithms of the inematical variables One may thus factorize the integrands to be left with a number of complex contour integrals These can in turn be evaluated by means of residue calculus Upon calculating these contour integrals, the result will be expressed as infinite series over one or more summation variables For typical applications to Quantum Field Theory, see e g [9] As a result of Feynman integrals, the Riemann zeta function [, ]: ζz appears frequently In particular, we shall need its values for and : π ζ , n 6 ζ 57 n n z 99 Mathematics Subject Classification Primary 4A5, 4B5; Secondary M99, B5, C, E, 8Q Key words and phrases Euler series, hypergeometric series, Riemann zeta function, psi function, polylogarithms, Clausen s function This research has been supported by the Research Council of Norway, and by DESY

2 ODD MAGNE OGREID AND PER OSLAND In Feynman integrals, these constants often appear as results of integrations where polylogarithms are involved The above mentioned method, using Mellin transforms, has been applied to Feynman integrals appearing in two-loop studies of Bhabha scattering [4, 5] In recent wor [9], the transcendental constants appear when summing series instead of resulting from integrations A large number of the series encountered are not found in the familiar tables This is the case for some of the one-dimensional series, and in particular for the twodimensional ones The purpose of this article is to present results for some of the series encountered These can be expressed in terms of a few constants, including ζ, ζ and the Catalan constant G In addition, the constant Cl π/ appears frequently It seems to us that few of these results are nown, in particular none of those involving Clausen s function Some basic properties of Clausen s function are given in Appendix A Similar results, including some of those given here, have been presented in [, 6, 7,,, ] An excellent article on triple Euler series can be found in [8] Here, the reader will also find an appendix written by D Broadhurst on the connection between Euler series, Quantum Field Theory and Knot Theory For further references on such series, we refer the reader to [4] Before we turn to the evaluation of these sums, we note that in this wor we will interchange integrations, differentiations, sums and limits at will In general, care should be exercised in doing so For the calculations shown here, all these interchanges are allowed Let us consider the family of series One-dimensional series n [γ + ψ + n] n j n j, where γ is the Euler constant, is a positive integer and ψz is the logarithmic derivative of the gamma function [see Appendix A] For, this reduces to one of the double Euler series Our family of series can all be expressed in terms of a rational multiple of ζ and a finite sum over Clausen s function [5] This is true also for the corresponding alternating series: Theorem n [γ + ψ + n] n + n [γ + ψ + n] 9 8 ζ + π j ζ + π j πj j Cl, πj j Cl + π, for,,,, where the sums over j are understood to be zero when For low values of, the sums over Clausen s function can be expressed in terms of Cl π/ and the Catalan constant G Explicit results are given in Appendix B

3 ONE- AND TWO-DIMENSIONAL SERIES Proof We start with the proof of By using the integral representation A of the psi function, we can rewrite the series as n [γ + ψ + n] n tn t ζ Li t, t where we have interchanged integration and summation, thus enabling us to express the sum as an integral by using A4 By the factorization formula A7, the argument of the dilogarithm can be linearized, yielding ζ j Li ω j t, t where ω e πi/ If one tries to calculate this integral term by term, one will see that each term is divergent although the sum converges For the purpose of splitting up the integral, we introduce a regulator as follows, ζ j Li ω j t lim t x lim x ζ j Li ω j t xt ζ xt j Li ω j t xt The first of these integrals is trivial, while for those under the sum we will need the result for I x, a Li at xt in the limit x This integral is studied in Appendix C Using the result from eq C4, we get lim x log x x ζ + { } log x Li ω j + S, ω j + Li ω j x j By using the factorization formula A7, we get lim x log x ζ + x ζ + S, ω j, j log x x Li + S, ω j + Li where the divergent parts cancel and we may let x To proceed further, we note that j S,ω j j S,ω j, since an inversion of the argument j

4 4 ODD MAGNE OGREID AND PER OSLAND simply corresponds to performing the sum in reverse order Thus, ζ + S, ω j ζ + ] ω [S, ω j + S, j j ζ + j j [ Li ω j ] 6 log ω j log ω j Li ω j + ζ, where we have used the identity A Next, we may use the factorization formula A7 and the fact that log ω j iπj/ This enables us to follow our convention that log iπ Thus, we get + ζ + j [ iπ 6 j j iπ Li ω j The result must be real Thus, we may drop all the imaginary parts which eventually will cancel, and we are left with + ζ + π j Im { Li ω j } j + ζ + π + j ζ + π j πj j Cl πj j Cl In the first step we have used the fact that on the unit circle, the imaginary part of the dilogarithm is Clausen s function, which vanishes when the argument is an integer multiple of π This completes the proof of the first part of the theorem For the alternating series we will use a similar procedure By using the integral representation A for the psi function and performing the sum over n, we are left with Li Li [φt ], t where φ We now follow the same procedure as in the proof of, except that we let the sum in the factorization formula run from to Hence, we arrive at Li + S, φω j Performing the sum in reverse order simply corresponds to the substitution ω j ω j Combining this with the fact that we could equally well have introduced φ instead of φ, we find that we will get 4 4 ζ + j j [ ] S, φω j + S, φω j ]

5 ONE- AND TWO-DIMENSIONAL SERIES 5 We use the fact that log φω j iπ[j+/ ], which preserves the convention log iπ, to get 9 ζ + 8 j [ iπ 6 j + ] j + iπ Li φω j Again, we now that the result must be real, and we drop all the imaginary parts Thus, 9 πj ζ + π j Cl 8 + π j This completes the proof of the second part of the theorem An immediate corollary of this theorem is: Corollary 5 6 [γ + ψn] n + n n [γ + ψn] 8 ζ + π j ζ + π j for,,,, where the sum over j vanishes for πj j Cl, πj j Cl + π, Proof This corollary follows immediately from the theorem by using the recurrence formula A for the psi function For certain low values of, the sums over Clausen s function may be simplified Thus, we may state exact and compact results for a considerable number of series Such results are collected in Appendix B In the Theorem, n appears in the denominator of the summand For higher powers of n, part of the same procedure can be carried out when summing the corresponding series However, for the higher powers, no simple result appears nown for the sum over Nielsen s functions on the unit circle We refer to Appendix D for more details We now turn to some other series Series 7 [Γn] n Γn 8 ζ + 4π π Cl Proof We start by using the duplication formula for the gamma function to get a hypergeometric series, [Γn] n Γn Γ Γn n n Γ + n F,,, ;,, ; F,, ; 4, ; 4 t

6 6 ODD MAGNE OGREID AND PER OSLAND In the last step we have used the integral representation for the function 4 F, eq A By using the identity 745 of [] and afterwards maing the substitution t sin u we get π t 4 t arcsin du tan u u We will use integration by parts to decompose this integral We get u log sin u π π 4 du u log sin u π 4 du u log sin u Invoing eqs 65 and 646 of [5], we get { 4 ζ π π π } Cl Cl 4π Cl π 8 ζ Series 9 Proof We may rewrite this series as n n [Γn] Γn n n [Γn] Γn 4 5 ζ n+ n [n!] n! 4 5 ζ, where eq in Chapter 9 of [] gives us the sum of this series Two-dimensional series A large number of two-dimensional series encountered in Feynman integral calculations can also be summed analytically Some results for such series will be presented here along with proofs, many of which are based on the results of our theorem The simpler ones are evaluated by summing over one variable and recognizing the result as a one-dimensional series for which we now the sum For the other series, we will need to use integration- or differentiation methods to analytically evaluate the sum When summing the simpler two-dimensional series, we will often find the following result useful, n + a n + b + a + b F, + a, + b; + a, + b; [ψ + b ψ + a], a b b a The last step follows by using eq 744 of [] There is a misprint in eq 65 of [5] The correct result is θ dθ log sin θ ζ θcl θ Cl θ

7 ONE- AND TWO-DIMENSIONAL SERIES 7 Another useful result is Γn + Γ + Γ + n + Γ + F, + ; + ; Γ Γ + This follows immediately by recognizing the series as a hypergeometric function and using A Series n + n + ζ Proof We start by summing over n, using, to get n + n + [γ + ψ + ] + [γ + ψ] [γ + ψ] ζ, where we have made use of B in the last step Series 4 4 n + n + 4 ζ Proof Also here we start by summing over n, using, to get n + n + [ψ + ψ + ] 4 ζ, where we have made use of B and B4 in the last step Series 5 5 n n n + n ζ Proof This follows as an immediate corollary from the previous result We rewrite the sum as n + n + + n + n + ζ + 4 ζ 5 4 ζ This result can also be found with a different proof in eq of [] by using the identity to relate these series Series 6 6 ΓΓn +! Γ + n + ζ

8 8 ODD MAGNE OGREID AND PER OSLAND Proof We start by summing over n, using, to get ΓΓn +! Γ + n + ζ Series 7 7 ΓΓn +! Γ + n + ζ Proof The sum over n is the same as above Thus, ΓΓn +! Γ + n + ζ Series 8 8 ΓΓn +! Γ + n + [γ + ψ] ζ Proof Again we start by summing over n, using, to get ΓΓn +! Γ + n + [γ + ψ] where we have made use of B in the last step Series 9 9! ΓΓn + [γ + ψ + ] ζ Γ + n + [γ + ψ] ζ, Proof This result follows as an immediate corollary of the two previous results by using the recurrence relation A Series! ΓΓn + [γ + ψ + ] Γ + n + 8 ζ Proof We start once more by summing over n, using, to get ΓΓn +! Γ + n + [γ + ψ + ] where we have made use of B4 in the last step [γ + ψ + ] 8 ζ, We now turn to similar series involving n in the argument of the psi function Here, the results and are not sufficient We will require the use of integration- and differentiation methods throughout the rest of this section

9 ONE- AND TWO-DIMENSIONAL SERIES 9 Series ΓΓn +! Γ + n + [γ + ψn] 7 8 ζ Proof The relation A is used to write ΓΓn +! Γ + n + [γ + ψn] t ΓΓn +! Γ + n + } { + F, + ; + ; t, tn t where we have now interchanged the order of integration and summation, thereby being able to perform the sum over n Next, we apply the integral representation A8 for F before summing over Thus, we get { t { ζ + t t t { ζ + { ζ + ds ts ds ts log [ ] s } ts [ s ts ]} ds [log s + log t s log ts] ts ds log s ts + ds log t s ts } } ds log ts ts Using eqs and 45 of [] along with eq A of [5] to perform the integration over s, we get { ζ [ Li t + t t t log t log t +Li t Li [ t ] ]} log t The identity of [] is used to rewrite this as t t { ζ + [ Li [ t ] + logtlog t ζ ]} t { ζ + [ Li t + logtlog t ζ ]} t t t logtlog t + [ Li t t ζ ], t t

10 ODD MAGNE OGREID AND PER OSLAND where we now have substituted t t in the second step By expanding in partial fractions, we arrive at logtlog t + logtlog t t t + [ Li t tζ ] + t logtlog t + t [ + Li t + Li t ζ] + ζ t [ Li t t ζ ] t t [Li t + Li t] ζ The first two integrals have been combined by substituting t t in the second one Identity 8 of [] has been used to transform the two last integrals In order to proceed, the first two of these integrals are evaluated using eqs 6, 8 and 85 of [] The result is 9 4 ζ + t [Li t ζ] + [ Li t + t ζ] 7 8 ζ, where eqs 89 and 8 of [] have been used in the last step Series ΓΓn +! Γ + n + [γ + ψn + ] 5 4 ζ Proof The recurrence relation A for the psi function is used to write ΓΓn + [γ + ψn + ]! Γ + n ζ + ΓΓn +! Γ + n + [γ + ψn] +! + F,, + ;, + ;, ΓΓn + Γ + n + n where now the sum over n has been performed in the last term By using eq 7444 of [], we get 7 8 ζ + {ψ + ψ + } 5 4 ζ, where in the last step, eqs B and B4 have been used Note: There is a sign error in eq 8 of [] The correct result is [ Li t + t ζ] 5 8 ζ

11 ONE- AND TWO-DIMENSIONAL SERIES Series Proof We can rewrite the sum as! ΓΓn +! Γ + n + [γ + ψn + ] ζ ΓΓn + [γ + ψn + ] Γ + n + ΓΓn + γ! Γ + n + + d dx γ + d dx x γ + d Γ + + xγ dx x! xγ + [ ] + γ + ψ + ζ + [γ + ψ + ] ζ, x where the result B has been used in the last step Series 4 4! Proof We can rewrite this series as! ΓΓn + + x! Γ + n + Γ + + xγ F, + + x; + ;!Γ + ΓΓn + 5 [γ + ψ + n + ] Γ + n + 8 ζ ΓΓn + [γ + ψ + n + ] Γ + n + ΓΓn + γ! Γ + n + + d ΓΓn + dx x! Γ + n + x γ + d Γ dx x Γ + x F, + ; + x; γ + d Γ dx x xγ + x [ ] + γ + ψ + ζ + [γ + ψ + ] 5 8 ζ, where the result B4 has been used in the last step

12 ODD MAGNE OGREID AND PER OSLAND Series 5 5 n n!! [Γn + ] n + Γn + 8 ζ 4π π Cl Proof First, we use the duplication formula for the gamma function and perform the sum over n to get a hypergeometric function, n n!! n [Γn + ] n + Γn + n!! ΓΓ n + Γ + F [Γn + ] Γ Γ + n + n+ 4,, ; +, + ; 4 4 By using the integral representation A9, we are able to perform the sum over, ds ds ds ds s t t st st t 4 st + st st t log st t log st 4 4 st 4 4 st Next, we integrate over s, and thereafter introduce the series representation of the dilogarithm: t t t Li 4 t t n t +n t n 4 ΓnΓ n Γ + n t n n n 8 4 ζ 4π π Cl n 4 In the last two steps, we performed the integration over t and arrived at a series encountered in the proof of Series Series 6 6 n n!! n + [Γn + ] Γn ζ 4π π Cl + π log + πg 4

13 ONE- AND TWO-DIMENSIONAL SERIES Proof The sum over n is the same as for the previous case, [Γn + ] n!! n + Γn + n ΓΓ Γ + F,, ; +, + ; 4 4 By using the integral representation A9 of F, we are able to perform the sum over, ds s t t 4 st 4 ds st st t 4 st ds st t log st 4 st { ds st log st log st } t 4 t t t Li t + 4 t t Li The first of these two integrals is nown from the previous proof, while the second one is calculated in the same way as we did for the first one Thus, we get 8 ζ 4π π Cl + t t n n t n 8 ζ 4π π n Cl + n t +n t 8 ζ 4π Cl 8 ζ 4π Cl 8 ζ 4π Cl π ΓnΓ + n Γ + n π + 4 F,,, ;,, ; π + F,, ;, ; t, where we have used the integral representation for 4 F Next, we use eq 745 of [] to get 8 ζ 4π π Cl t + 4 t arcsin t The substitution sin w yields 8 ζ 4π π Cl + π dw tan w w

14 4 ODD MAGNE OGREID AND PER OSLAND To evaluate this integral, we integrate by parts to get 8 ζ 4π π [ Cl + w log sin w ] π 8 ζ 4π π π Cl + π 4 log 4 dw w log 8 ζ 4π π Cl + π 5 log ζ + πg ζ 4π Cl π + π 4 by using eq 65 of [5] in the last step log + πg, π [ 4 dw w log sin w ] [ sin w ] Series 7 7! nn + Γn + Γn + Γn + 47 ζ 4π π Cl To be able to prove this result, we will need the following lemma: Lemma 8 Γ n n Γ + n Γ [ψ ψ] Γ + Proof of the lemma First, we rewrite the series as 9 lim y Γ + y n Γ n yγ + y Γ + n The combination of gamma functions inside the sum is a beta function which we will represent as an integral, Γ n yγ + y Γ + n t + n y t +y, where now < Re y < n for the integral to converge The factor /n is also represented as an integral, n dx x n

15 ONE- AND TWO-DIMENSIONAL SERIES 5 The sum in 9 can then be rewritten as Γ n yγ + y n Γ + n dx x n t + n y t +y dx dx t + y t +y t + y t +y [ x n t [ dx x t + y t +y x/t x/t ] t/x t/x For the purpose of integrating over t, we would lie to split the integral into two parts, according to the two terms in the numerator In order to do so, avoiding the singularity from the denominator, we introduce a shift iǫ in the denominator Thus, [ ] dx x t + y t +y t/x t/x + iǫ { dx x Γ yγ + y F, y; ; x iǫ Γ yγ + y x F, y; + ; } Γ + x iǫ The two hypergeometric functions may be combined using the transformation formula 76 of [] with a, b, c + y and z x + iǫ Thus, we may let ǫ, to get Γ + yγ y dx F, ; + y; x Γ Γ + yγ y F,, ;, + y; Γ Γ + yγ y [ψ + y ψ + y] Γ + Γ + yγ y { ψ y ψ y Γ + + π cot[π y] π cot[π y] } Γ + yγ y Γ + [ψ y ψ y], where in the first two steps we have used eqs 79 and 7444 of [] We have also used the reflection formula, eq 67 of [], for the psi function Thus, ]

16 6 ODD MAGNE OGREID AND PER OSLAND we have shown that Γ n y Γ y [ψ y ψ y], n Γ + n Γ + valid for < Re y < By analytic continuation, the result can be extended to all values of y, except y, +, +, By letting y, we find that Γ n n Γ + n Γ [ψ ψ], Γ + and the lemma has been proven Proof of the series First, we rewrite the series as! nn +! nn + Γn + Γn + Γn + Γn + Γn + Γn + [Γ] Γ The second of these two series is already nown cf Series, whereas for the two-dimensional one, we will rewrite it using the identity a n, a n, n, which is easily derived from identities given in chapter 4 of [] We get n By the lemma, this equals ΓΓ n ΓΓ + n 4π π Cl + 8 ζ [ψ ψ] 4π π Cl ζ ζ 4π π Cl, where we have used B and B Thus, the proof is complete It is interesting to note that the sums that appear in moderate-order Feynman integral calculations apparently can all be expressed in terms of nown constants This suggests that such integrals in some sense are of limited complexity For a qualification of this statement, we refer to the appendix of [8] Appendix A Some special functions and identities Below, we collect some definitions and properties of special functions that are frequently used in the proofs

17 ONE- AND TWO-DIMENSIONAL SERIES 7 The psi function The psi function is defined as the logarithmic derivative of the gamma function, ψz d/dz log Γz Γ z/γz, and has the following integral representation cf eq 6 of [], A γ + ψz tz, t where γ is Euler s constant, and ψ γ From the integral representation, one immediately finds that in the case of positive integer arguments, we have A γ + ψn n The psi function satisfies the following recurrence relation see eq 65 of [], j j A ψ + z ψz + z Polylogarithms Each of the polylogarithm functions can be represented as a series, z A4 Li n z, z <, n,,,, n or as an integral, A5 A6 Li n z n n! z logn tlog tz, n,, 4, t Li n t, n,,,, t with Li n ζn, n,, 4,, and Li n / n ζn, n,, 4, The following factorization formula eq 74 of [5] is valid for the polylogarithms, A7 Li n t n Li n ω j t, n,,,, j where ω e πi/ These and further properties of the polylogarithms and related functions can be found in [5, ] Nielsen s generalized polylogarithms The functions S n,p z are Nielsen s generalized polylogarithms We will mae use of these functions for p, in which case they are given by the following integral representations [], A8 A9 S n, z n n! z logn tlog zt, n,,, t S n,t, n,,,, t where it is understood that S, z log z We shall be needing the following identity for S, z, A S, z + S, Li z z 6 log z log zli z + ζ,

18 8 ODD MAGNE OGREID AND PER OSLAND given in eq 5 of [] We will use the convention log iπ Thus, the formula is seen to be valid also for z Further identities regarding the functions S n,p z can be found in [] Nielsen s original wor is found in [7] Clausen s functions Each of Clausen s functions can be represented as a series see [5], sinθ Cl n θ n, A cosθ Cl n θ n, where n can be any positive integer Clausen s functions are seen to be periodic with period π We shall in this article need only the functions Cl θ, Cl θ and Cl θ From the definition, we see that the function Cl θ is antisymmetric, Cl θ Cl θ When the argument is an integer multiple of π, the function vanishes, Cl π On the unit circle, the imaginary part of the dilogarithm is Clausen s function Cl θ, A Im { Li e iθ } Cl θ This function satisfies the factorization formula A Cl θ Cl θ Cl π θ The Clausen function Cl θ has its maximum value for θ π/, Cl π/ 4 94 Furthermore, π Cl π π A4 Cl, Cl G, where G is the Catalan constant Hypergeometric functions We will frequently use the series definitions of the following hypergeometric functions, a n b n z n A5 F a, b; c; z c n n!, A6 A7 F a, b, c; d, e; z 4F a, b, c, d; e, f, g; z where a n is the Pochhammer symbol, a n Γa + n Γa n n n a n b n c n d n e n z n n!, a n b n c n d n e n f n g n aa + a + n It is evident that these functions are all symmetric with respect to the numerator or the denominator arguments All these series converge for z < Conditions for convergence on the unit circle are given in Chapter 7 of [] z n n!,

19 ONE- AND TWO-DIMENSIONAL SERIES 9 These functions all have various integral representations We shall be using the following ones, A8 A9 A A F a, b; c; z F a, b, c; d, e; z 4F a, b, c, d; e, f, g; z Γc ΓbΓc b ΓdΓe ΓaΓbΓd aγe b t b t c b tz a, t a t b t d a t e b t t z c Γe ΓcΓe c Γg Γg d t c t e c F a, b; d; zt, t d t g d F a, b, c; e, f; zt Conditions for convergence of these integrals are also given in Chapter 7 of [] In the case when z, we get A F a, b; c; ΓcΓc a b Γc aγc b Appendix B Explicit results for one-dimensional series The finite sums in the Theorem and Corollary of may be further simplified for small values of One may then use properties of the Clausen function given in Appendix A to reduce the sums to a single Clausen function Here, we collect the results for such series involving the psi function: B B B B4 B5 B6 B7 B8 [γ + ψn] ζ n [γ + ψ + n] ζ n n [γ + ψn] 9 4 ζ [γ + ψ + n] n 4 ζ 4 [γ + ψn] n ζ π π Cl n [γ + ψ + n] 5ζ π π Cl 65 [γ + ψ4n] ζ πg n 8 67 [γ + ψ + 4n] ζ πg n 8

20 ODD MAGNE OGREID AND PER OSLAND B9 B B B B B4 B5 B6 n n 7 [γ + ψ6n] 7 [γ + ψ + 6n] 4 n [γ + ψn] 8 ζ n 6π π ζ Cl n [γ + ψ + n] 5 8 ζ n 6π π ζ Cl n n [γ + ψn] 9 ζ πg 6 n n [γ + ψ + n] ζ πg 6 n n [γ + ψn] 5 π 8 ζ πcl n n [γ + ψ + n] π 8 ζ πcl At least four of these results were probably nown by Euler, in particular the results of eqs B, B, B and B A review of the history of this type of series, nown as Euler series, can be found in [] Appendix C The integral I p x, a In the proof of the theorem in, we will need a result for the integral in the limit x, which is I p x, a Li pat C xt for p This integral is encountered for arbitrary positive integers p when studying the generalization of our theorem in Appendix D This section will be devoted to the study of this integral in the appropriate limit x Integrating by parts, we obtain I p x, a x log xli pa + x t log xtli p at Here, the singular part as x has been isolated, and we may set x in the remaining integral to get I p x, a x log xli pa Λ p a, where C Λ ν a t Here, we will need the following result C log tli ν at Λ ν a S ν, a + Li ν+ a,

21 ONE- AND TWO-DIMENSIONAL SERIES which can be proved by induction By noting that Li z z/ z, we find from eqs 7 and 5 of [] that the result holds for ν Now, suppose the result is valid for ν Consider Λ ν a t log tli ν at t dy log t y Li ν ayt, which follows by eq 7 of [] By interchanging the order of integration and invoing our assumption, we may perform the t integration to get Λ ν a dy y [S ν,ay + Li ν+ ay] S ν, a + Li ν+ a by once again using eq 7 of [] By induction, the result C holds for all non-negative integers ν Thus, we conclude that C4 I p x, a x log xli pa S p, a Li p+ a in the limit x Appendix D Generalizations One may foresee the need for generalizations of the results of the theorem, and, to powers n p instead of n Let us therefore consider the sum D σ p n p [γ + ψ + n] n j n p j, where p,, 4, and,,, Introducing the integral representation for the psi function, and the regulator lie in, we find σ p n p tn lim ζp Li pt t x xt For the polylogarithm, we use the factorization formula A7 to get σ p lim x log xζp p I p x, ω j, x where we have introduced the integral I p x, a Li pat xt This integral was calculated in Appendix C Using this result and invoing the factorization formula A7, we note that the singular parts cancel Thus, the result is [γ + ψ + n] p np j [ Sp, ω j + Li p+ ω j ] j ζp + + p S p, ω j j

22 ODD MAGNE OGREID AND PER OSLAND As in the case with p, it is only the real part of the Nielsen function that enters, S m, ω j + S m, ω j We have not been able to find any simple expression for this real part for m Thus, we leave the result as n p [γ + ψ + n] D ζp + + p S p, ω j Similarly, for the generalization of the alternating series, we find D n n p [γ + ψ + n] j p ζp + + p S p, ω j+/ j When p, only the alternating series converges We have been able to prove the following result, D4 n [γ + ψ + n] n 4 + ζ + and the immediate corollary thereof, n [γ + ψn] D5 n 4 ζ + j j { πj Cl + π }, { πj Cl + π } Proof By applying D we find that n [γ + ψ + n] n ζ + S, ω j+/ ζ + j { } Li ω j+/ by maing use of the fact that S, z {Li z} On the unit circle, we now from [5] that Li θ Cl θ + igl θ By retaining only the real parts of the expression, we arrive at ζ + j j j { [ πj Cl + π ] [ πj Gl + π ] } The function Gl θ is a well-nown Fourier series equal to the π-periodic extension of π θ, < θ < π This fact is used to perform the finite sum over the part containing Gl θ, π ζ j 8 + { πj Cl + π } 4 + ζ + j j { πj Cl + π }

23 ONE- AND TWO-DIMENSIONAL SERIES By using the fact that Cl θ log sin θ, we may now state the following results: D6 n [γ + ψn] n log D7 n [γ + ψ + n] n ζ + log D8 n [γ + ψn] n 8 ζ + 4 log D9 n [γ + ψ + n] 5 n 8 ζ + 4 log D n [γ + ψn] n ζ + log D n [γ + ψ + n] 5 n 6 ζ + log For higher values of, the sums do not turn out to be this simple

24 4 ODD MAGNE OGREID AND PER OSLAND References [] M Abramowitz and I A Stegun, Handboo of Mathematical Functions, Tenth printing National Bureau of Standards Applied Mathematics Series no 55, Washington, 97 [] L V Ahlfors, Complex Analysis, Third edition McGraw-Hill Inc, New Yor, 979 [] B C Bern, Ramanujan s Noteboos Part Springer-Verlag, New Yor, 985, Chapter 9 [4] H J Bhabha, Proc Roy Soc A [5] K S Bjørevoll, G Fäl and P Osland, Nucl Phys B ; Nucl Phys B [6] D Borwein and JM Borwein, Proceedings of the American Mathematical Society Vol, Number [7] D Borwein, JM Borwein and R Girgensohn, Proceedings of the Edinburgh Mathematical Society [8] JM Borwein and R Girgensohn, Electronic J Combinatorics 996 R [9] H Cheng and T T Wu, Expanding Protons: Scattering at High Energies, The MIT Press, Cambridge, Massachusetts, 987 [] PJ De Doelder, Journal of Computational and Applied Mathematics [] A Devoto and D W Due, La Rivista del Nuovo Cimento vol 7, no [] R P Feynman, Phys Rev [] E R Hansen, A Table of Series and Products Prentice-Hall, New Jersey, 975 [4] M Hoffman has compiled a list of references on multiple harmonic series and Euler sums at his website, [5] L Lewin, Polylogarithms and Associated Functions North Holland Publishing, New Yor, 98 [6] T Myint-U and L Debnath, Partial Differential Equations for Scientists and Engineers, Third Edition Prentice-Hall, New Jersey, 987, p 6 [7] N Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen, Nova Acta, Abh der Kaiserl Leop-Carol Deutschen Aademie der Naturforscher, Band XC 99 [8] F Oberhettinger, Tables of Mellin Transforms Springer-Verlag, Berlin-Heidelberg-New Yor, 974 [9] O M Ogreid, PhD thesis, unpublished [] A P Prudniov, Yu A Brychov and O I Marichev, Integrals and Series, bd Gordon and Breach, New Yor, 99 [] L-C Shen, Transactions of the American Mathematical Society Vol 47, Number [] R Sitaramachandrarao and A Sivaramasarma, Indian J pure appl Math, Department of Physics, University of Bergen, Allégt 55, N-57 Bergen, Norway address: OddOgreid@fiuibno URL: Department of Physics, University of Bergen, Allégt 55, N-57 Bergen, Norway Current address: Deutsches Eletronen-Synchrotron DESY, D-6 Hamburg, Germany address: PerOsland@fiuibno URL: