Special Values of Generalized Log-sine Integrals

Special V alues of Generalized Log-sine Integrals Jonathan M. Borwe in Unive rsity of Ne wcastle Callaghan, NSW 2308, Austr alia jonathan .borwein @ne wcastle.ed u.au Armin Strau b T ulane Univ ersity Ne w Orleans, LA 70118, USA astraub @tulane .edu ABSTRA CT W e study generalize d log-sine integral s at sp ecial v alues. At π and multiples th ereof exp licit ev aluatio ns are obtained in terms of Niel sen polylogarithms at ± 1. F or g eneral argu- ments we present algorithmic ev al uations i nv olving Nielsen p olylogari thms at related argumen ts. In particular, we con- sider log-sine integral s at π / 3 which ev aluate in terms of p olylogari thms at the sixth root of unity . An implemen ta- tion of our results for the compu ter algebra systems Mathe- matic a and S AG E is provided. Categories and Subject Descriptors I.1.1 [ Symbolic and Algebraic Manipulation ]: Expres- sions and Their Representatio n; I.1.2 [ Symboli c an d Al- gebraic Manipulation ]: Algorithms General T erms Algorithms, Theory Keyw ords log-sine integ rals, multiple p olylogari thms, multiple zeta v al- ues, Clausen functions 1. INTR ODUCTION F or n = 1 , 2 , . . . and k ≥ 0, w e consider the (generalized) lo g- sine inte gr als defined by Ls ( k ) n ( σ ) := − Z σ 0 θ k log n − 1 − k     2 sin θ 2     d θ . (1) The mod ulus is not needed for 0 ≤ σ ≤ 2 π . F or k = 0 these are the (b asic) log-sine in tegrals Ls n ( σ ) := Ls (0) n ( σ ). V arious log-sine in tegral ev aluatio ns may b e found in [ 20 , § 7.6 & § 7.9]. In this p aper, we will b e concerned with ev aluations of the log-sine in tegrals Ls ( k ) n ( σ ) for special v alues of σ . Suc h Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not m ade or distrib ute d for profit or commercia l adv antage and that copies bear this notice and the full cita tion on the first page. T o co py otherwise , to republi sh, to post on servers or to redistrib ute to l ists, requi res prior speci fic permission and/or a fee. ISSA C’11, June 8–11, 2011, San Jose, California , US A. Copyri ght 2011 ACM 978 -1-4503-0675 -1/11/06 ...$10.00. ev aluatio ns are u seful for physics [ 15 ]: log-sine integrals ap- p eared for instance in recent work on the ε - expansion of v ario us F eynman d iagrams in the calculation of h igher terms in the ε -expansion, [ 8 , 16 , 9 , 11 , 14 ]. Of particular impor- tance are the log-sine integrals at th e sp ecial v alues π / 3, π / 2, 2 π / 3, π . The log-sine integrals also app ear in many settings in num b er theory and analysis: classes of inv erse binomial sums can be expressed in terms of generalized log- sine integ rals, [ 10 , 4 ]. In Section 3 we focu s on ev aluatio ns of log-sine and related integ rals at π . General arguments are considered in Section 5 with a fo cus on the case π / 3 in S ection 5.1 . Imaginary arguments are briefly discussed in 5.2 . The results obtained are suitable for implemen tation in a computer alge bra sys- tem. Such an implementati on is pro vided f or Mathematic a and SAGE, and is d escribed in Section 7 . This complements existing pac k ages such as lsjk [ 15 ] for numerical ev aluations of log-sine integrals or HPL [ 21 ] as well as [ 25 ] for working with multiple p olylogarithms. F urther motiv ation for such eva luations was sparked by our recent study [ 6 ] of certain m ultiple Mahler m e asur es . F or k functions (typically Laurent p olynomials) in n v ariables the multiple Mahler measure µ ( P 1 , P 2 , . . . , P k ), introduced in [ 18 ], is defined b y Z 1 0 · · · Z 1 0 k Y j =1 log    P j  e 2 πit 1 , . . . , e 2 πit n     d t 1 d t 2 . . . d t n . When P = P 1 = P 2 = · · · = P k this devolv es to a higher Mahler me asur e , µ k ( P ), as introduced an d examined in [ 18 ]. When k = 1 b oth reduce t o the standard (logarithmic) Mahler me asur e [ 7 ]. The multiple Mahler measure µ k (1 + x + y ∗ ) := µ (1 + x + y 1 , 1 + x + y 2 , . . . , 1 + x + y k ) (2) w as studied by Sasaki [ 24 , Lemma 1] who p ro vided an ev al- uation of µ 2 (1 + x + y ∗ ). It was observ ed in [ 6 ] that µ k (1 + x + y ∗ ) = 1 π Ls k +1  π 3  − 1 π Ls k +1 ( π ) . (3) Man y other Mahler measures studied in [ 6 , 1 ] were shown to h a ve ev aluations inv olving generalized log-sine integrals at π and π / 3 as wel l. T o our knowledge, this is the most ex acting such study undertaken — perhaps because it w ould be quite impossible without mod ern computational to ols and absent a use of the quite recent understanding of multiple p olylogari thms and multiple zeta v alues [ 3 ]. 2. PRELIMIN ARIES In th e follo wing, w e will d en ote the m ultiple p olyl o garithm as studied for instance in [ 4 ] and [ 2 , Ch. 3] by Li a 1 ,...,a k ( z ) := X n 1 > ··· >n k > 0 z n 1 n a 1 1 · · · n a k k . F or our purp oses, the a 1 , . . . , a k will usually b e p ositiv e in- tegers and a 1 ≥ 2 so that th e sum conv erges for all | z | ≤ 1. F or example, Li 2 , 1 ( z ) = P ∞ k =1 z k k 2 P k − 1 j =1 1 j . In particular, Li k ( x ) := P ∞ n =1 x n n k is the p olylo garith m of or der k . The usual notation will b e used for rep etitions so that, for in- stance, Li 2 , { 1 } 3 ( z ) = Li 2 , 1 , 1 , 1 ( z ). Moreo v er, multiple zeta values are denoted by ζ ( a 1 , . . . , a k ) := Li a 1 ,...,a k (1) . Similarly , w e consider the multiple Clausen functions (C l) and multiple Glai sher functions (Gl) of depth k and weigh t w = a 1 + . . . + a k defined as Cl a 1 ,...,a k ( θ ) =  Im Li a 1 ,...,a k ( e iθ ) if w even Re Li a 1 ,...,a k ( e iθ ) if w o dd  , (4) Gl a 1 ,...,a k ( θ ) =  Re Li a 1 ,...,a k ( e iθ ) if w even Im Li a 1 ,...,a k ( e iθ ) if w o dd  , (5) in accordance with [ 20 ]. Of particular imp ortance will b e the case of θ = π / 3 which h as also b een considered in [ 4 ]. Our other n otation and usage is largely consistent with that in [ 20 ] and that in the newly pub lished [ 23 ] in whic h most of th e requisite material is describ ed. Finally , a recen t elaboration of what is meant when we sp eak ab out ev alua- tions and “closed forms” is to b e found in [ 5 ]. 3. EV ALU A TIONS A T π 3.1 Basic log-sine integrals at π The exp onential generating fun ction, [ 19 , 20 ], − 1 π ∞ X m =0 Ls m +1 ( π ) λ m m ! = Γ (1 + λ ) Γ 2  1 + λ 2  = λ λ 2 ! (6) is w ell-kno wn and implies the recurrence ( − 1) n n ! Ls n +2 ( π ) = π α ( n + 1) + n − 2 X k =1 ( − 1) k ( k + 1)! α ( n − k ) Ls k +2 ( π ) , (7) where α ( m ) = (1 − 2 1 − m ) ζ ( m ). Example 1. (V alues of Ls n ( π )) W e ha ve Ls 2 ( π ) = 0 and − Ls 3 ( π ) = 1 12 π 3 Ls 4 ( π ) = 3 2 π ζ (3) − Ls 5 ( π ) = 19 240 π 5 Ls 6 ( π ) = 45 2 π ζ (5) + 5 4 π 3 ζ (3) − Ls 7 ( π ) = 275 1344 π 7 + 45 2 π ζ (3) 2 Ls 8 ( π ) = 2835 4 π ζ (7) + 315 8 π 3 ζ (5) + 133 32 π 5 ζ (3) , and so fo rth. The fact that each integral is a multiv ari- able rational p olynomial in π and zeta v alues follo ws directly from the recursion ( 7 ). Alternatively , these v alues may b e conv enien tly obtained from ( 6 ) by a computer algebra sy s- tem. F or instance, in Mathematic a the cod e FullSimpli fy[D[-Bino mial[x,x/2 ], {x,6}] /.x- >0] prod uces th e ab ov e eva luation of Ls 6 ( π ). ✸ 3.2 The log-sine-cosine integrals The log-sine-cosine integrals Lsc m,n ( σ ) := − Z σ 0 log m − 1     2 sin θ 2     log n − 1     2 cos θ 2     d θ (8) app ear in physical ap p lications as well, see for instance [ 9 , 14 ]. They h ave also b een considered by Lewin, [ 19 , 20 ], and he demonstrates h o w their v alues at σ = π may be obtained muc h the same as those of the log-sine integral s in Section 3.1 . As observ ed in [ 1 ], Lewin’s result can b e pu t in the form − 1 π ∞ X m,n =0 Lsc m +1 ,n +1 ( π ) x m m ! y n n ! = 2 x + y π Γ  1+ x 2  Γ  1+ y 2  Γ  1 + x + y 2  = x x/ 2 ! y y / 2 ! Γ  1 + x 2  Γ  1 + y 2  Γ  1 + x + y 2  . (9) The last form mak es it clear that th is is an ext ension of ( 6 ). The n otation Lsc h as b een introduced in [ 9 ] where ev alu- ations for other v al ues of σ and lo w wei ght can b e found. 3.3 Log-sine integrals at π As Lewin [ 20 , § 7.9] sketc hes, at least for small val ues of n an d k , the generalized log-sine integral s Ls ( k ) n ( π ) h a ve closed forms invo lving zeta val ues and K ummer-type con- stants such as Li 4 (1 / 2). This will b e made more precise in Remark 1 . Our analysis starts with the generating function identit y − X n,k ≥ 0 Ls ( k ) n + k +1 ( π ) λ n n ! ( iµ ) k k ! = Z π 0  2 sin θ 2  λ e iµθ d θ = i e iπ λ 2 B 1  µ − λ 2 , 1 + λ  − i e iπµ B 1 / 2  µ − λ 2 , − µ − λ 2  (10) giv en in [ 20 ]. Here B x is the inc omplete Beta fun ction: B x ( a, b ) = Z x 0 t a − 1 (1 − t ) b − 1 d t. W e shall show that with care — b ecause of the singularities at zero — ( 10 ) can b e differentia ted as needed as suggested by Lewin. Using t h e identities, val id for a, b > 0 and 0 < x < 1, B x ( a, b ) = x a (1 − x ) b − 1 a 2 F 1  1 − b, 1 a + 1     x x − 1  = x a (1 − x ) b a 2 F 1  a + b, 1 a + 1     x  , found for instance in [ 23 , § 8.17(ii)], th e generating fun ction ( 10 ) can b e rewritten as i e iπ λ 2  B 1  µ − λ 2 , 1 + λ  − B − 1  µ − λ 2 , 1 + λ  . Up on exp anding this we obtain th e follo wing computation- ally more accessible generating fun ction for Ls ( k ) n + k +1 ( π ): Theorem 1. F or 2 | µ | < λ < 1 we have − X n,k ≥ 0 Ls ( k ) n + k +1 ( π ) λ n n ! ( iµ ) k k ! = i X n ≥ 0 λ n ! ( − 1) n e iπ λ 2 − e iπµ µ − λ 2 + n . (11) W e now show how the log-sine integral s Ls ( k ) n ( π ) can quite comfortably be extracted from ( 11 ) b y differentiating its righ t-hand side. The case n = 0 is cov ered by: Pr oposition 1. We have d k d µ k d m d λ m i e iπ λ 2 − e iπµ µ − λ 2     λ =0 µ =0 = π 2 m ( iπ ) m + k B ( m + 1 , k + 1) . Pr oof. This may b e deduced from e x − e y x − y = X m,k ≥ 0 x m y k ( k + m + 1)! = X m,k ≥ 0 B ( m + 1 , k + 1) x m m ! y k k ! up on setting x = iπ λ/ 2 an d y = iπ µ . The next prop osition is most helpful in differentiation of the righ t-hand side of ( 11 ) for n ≥ 1, Here, we denote a multiple harmonic num b er by H [ α ] n − 1 := X n>i 1 >i 2 >...>i α 1 i 1 i 2 · · · i α . (12) If α = 0 we set H [0] n − 1 := 1. Pr oposition 2. F or n ≥ 1 ( − 1) α α !  d d λ  α λ n !     λ =0 = ( − 1) n n H [ α − 1] n − 1 . (13) Note that, for α ≥ 0, X n ≥ 0 ( ± 1) n n β H [ α ] n − 1 = Li β , { 1 } α ( ± 1) whic h shows that the ev aluation of the log-sine integ rals will inv olv e N ielsen p olylogarithms at ± 1, that is polylogarithms of the type Li a, { 1 } b ( ± 1). Using the Leibniz rule coupled with Prop osition 2 to dif- feren tiate ( 11 ) for n ≥ 1 and Prop osition 1 in the case n = 0, it is p ossible to exp licitly write Ls ( k ) n ( π ) as a finite su m of Nielsen p olylogarithms with co efficien ts only b eing rational multiples of p ow ers of π . The process is no w ex emplified for Ls (2) 4 ( π ) and Ls (1) 5 ( π ). Example 2. (Ls (2) 4 ( π )) T o find Ls (2) 4 ( π ) w e differentiate ( 11 ) once with resp ect to λ an d twice with resp ect to µ . T o simplify computation, we exploit the fact that the result will b e real which allows us to neglect imaginary parts: − Ls (2) 4 ( π ) = d 2 d µ 2 d d λ i X n ≥ 0 λ n ! ( − 1) n e iπ λ 2 − e iπµ µ − λ 2 + n     λ = µ =0 = 2 π X n ≥ 1 ( − 1) n +1 n 3 = 3 2 π ζ (3) . In th e second step w e were able to drop the term correspond - ing to n = 0 b ecause its con tribution − iπ 4 / 24 is purely imaginary as follow s a priori from Prop osition 2 . ✸ Example 3. (Ls (1) 5 ( π )) Similarly , setting Li ± a 1 ,...,a n := Li a 1 ,...,a n (1) − Li a 1 ,...,a n ( − 1) w e obtain Ls (1) 5 ( π ) as − Ls (1) 5 ( π ) = 3 4 X n ≥ 1 8(1 − ( − 1) n ) n 4  nH [2] n − 1 − H n − 1  + 6(1 − ( − 1) n ) n 5 − π 2 n 3 = 6 Li ± 3 , 1 , 1 − 6 Li ± 4 , 1 + 9 2 Li ± 5 − 3 4 π 2 ζ (3) = − 6 Li 3 , 1 , 1 ( − 1) + 105 32 ζ (5) − 1 4 π 2 ζ (3) . The last form is what is automatically produced by our pro- gram, see Example 13 , and is obtained from the prev ious expression by reducing t he p olylogarithms as discussed in Section 6 . ✸ The next example hints at the rapid ly gro wing complexity of these integral s, esp ecially when compared to the eva lua- tions giv en in Examples 2 and 3 . Example 4. (Ls (1) 6 ( π )) Proceeding as b efore we fi n d − Ls (1) 6 ( π ) = − 24 Li ± 3 , 1 , 1 , 1 +24 Li ± 4 , 1 , 1 − 18 Li ± 5 , 1 +12 Li ± 6 + 3 π 2 ζ (3 , 1) − 3 π 2 ζ (4) + π 6 480 = 24 L i 3 , 1 , 1 , 1 ( − 1) − 18 Li 5 , 1 ( − 1) + 3 ζ (3) 2 − 3 1120 π 6 . (14) In the first equalit y , the term π 6 / 480 is t he one corresp on d - ing to n = 0 in ( 11 ) ob t ained from Prop osition 1 . The second form is again the au t omaticall y reduced output of our program. ✸ R emark 1. F rom the form of ( 11 ) and ( 13 ) we fi nd that the log-sine integrals Ls ( k ) n ( π ) can b e expressed in terms of π and Nielsen polylogarithms at ± 1. Using the duality results in [ 3 , § 6.3, and Examp le 2.4] the p olylogarithms at − 1 ma y b e exp licitly reexpressed as multiple p olylogarithms at 1 / 2. Some examples are given in [ 6 ]. P articular cases of Theorem 1 hav e b een considered in [ 15 ] where explicit form ulae are given for Ls ( k ) n ( π ) where k = 0 , 1 , 2. ✸ 3.4 Log-sine integrals at 2 π As observed by Lewin [ 20 , 7.9.8], log-sine integrals at 2 π are expressible in terms of zeta v alues only . If we p roceed as in the case of ev aluations at π in ( 10 ) w e fi nd th at the re- sulting in tegral no w becomes expressible in terms of gamma functions: − X n,k ≥ 0 Ls ( k ) n + k +1 (2 π ) λ n n ! ( iµ ) k k ! = Z 2 π 0  2 sin θ 2  λ e iµθ d θ = 2 π e iµπ λ λ 2 + µ ! (15) The sp ecial case µ = 0, in the light of ( 20 ) which gives Ls n (2 π ) = 2 Ls n ( π ), reco v ers ( 6 ). W e ma y now extract log-sine integrals Ls ( k ) n (2 π ) in a sim- ilar wa y as describ ed in Section 3.1 . Example 5. F or instance, Ls (2) 5 (2 π ) = − 13 45 π 5 . W e remark that th is ev aluatio n is incorrectly given in [ 20 , (7.144)] as 7 π 5 / 30 u nderscoring an adva ntag e of automated ev aluatio ns ov er tables (indeed, t here are more misprints in [ 20 ] p oin ted out for instance in [ 9 , 15 ]). ✸ 3.5 Log-sine-polylog integrals Motiv ated by the integrals LsLsc k,i,j defined in [ 14 ] we sho w that the considerations of Section 3.3 can b e extended to more inv olv ed integrals including Ls ( k ) n ( π ; d ) := − Z π 0 θ k log n − k − 1  2 sin θ 2  Li d (e iθ ) d θ . On expressing Li d (e iθ ) as a series, rearranging, and apply ing Theorem 1 , w e obtain the follo wi ng exp onential generating function for Ls ( k ) n ( π ; d ): Cor ollar y 1. F or d ≥ 0 we have − X n,k ≥ 0 Ls ( k ) n + k +1 ( π ; d ) λ n n ! ( iµ ) k k ! = i X n ≥ 1 H n,d ( λ ) e iπ λ 2 − ( − 1) n e iπµ µ − λ 2 + n (16) wher e H n,d ( λ ) := n − 1 X k =0 ( − 1) k  λ k  ( n − k ) d . (17) W e note for 0 ≤ θ ≤ π th at Li − 1 (e iθ ) = − 1 /  2 sin θ 2  2 , Li 0 (e iθ ) = − 1 2 + i 2 cot θ 2 , while Li 1 (e iθ ) = − log  2 sin θ 2  + i π − θ 2 , and Li 2 (e iθ ) = ζ (2) + θ 2  θ 2 − π  + i Cl 2 ( θ ). R emark 2. Correspond ing results for an arbitrary Dirich- let series L a ,d ( x ) := P n ≥ 1 a n x n /n d can b e easily derived in the same fashion. I ndeed, for Ls ( k ) n ( π ; a , d ) := − Z π 0 θ k log n − k − 1  2 sin θ 2  L a ,d (e iθ ) d θ one derives the exp onential generating function ( 16 ) with H n,d ( λ ) replaced by H n, a ,d ( λ ) := n − 1 X k =0 ( − 1) k  λ k  a n − k ( n − k ) d . (18) This allo ws for Ls ( k ) n ( π ; a , d ) to b e extracted for many num- b er theoretic funct ions. I t do es not how ev er seem to cov er any of th e va lues of the LsLsc k,i,j function defined in [ 14 ] that are not already cove red by Corollary 1 . ✸ 4. QU ASIPERIODIC PROPER TIES As shown in [ 20 , (7.1.24)], it follo ws from the p eriod icity of the integra nd that, for intege rs m , Ls ( k ) n (2 mπ ) − Ls ( k ) n (2 mπ − σ ) = k X j =0 ( − 1) k − j (2 mπ ) j k j ! Ls ( k − j ) n − j ( σ ) . (19) Based on this q uasiperio dic prop erty of th e log-sine inte- grals, the results of Section 3.4 easily generalize to sh ow that log-sine integrals at multiples of 2 π ev aluate in t erms of zeta v alues. This is shown in Section 4.1 . It then follo ws from ( 19 ) th at log-sine integra ls at general arguments can b e reduced to log-sine integrals at arguments 0 ≤ σ ≤ π . This is discussed briefly in S ection 4.2 . Example 6. In the case k = 0, we ha ve that Ls n (2 mπ ) = 2 m Ls n ( π ) . ( 20) F or k = 1, sp ecializing ( 19 ) to σ = 2 mπ then yields Ls (1) n (2 mπ ) = 2 m 2 π Ls n − 1 ( π ) as is giv en in [ 20 , (7.1.23)]. ✸ 4.1 Log-sine integrals at multiples of 2 π F or o dd k , sp ecializing ( 19 ) to σ = 2 mπ , w e find 2 Ls ( k ) n (2 mπ ) = k X j =1 ( − 1) j − 1 (2 mπ ) j k j ! Ls ( k − j ) n − j (2 mπ ) giving Ls ( k ) n (2 mπ ) in t erms of low er order log-sine integrals. More generally , on setting σ = 2 π in ( 19 ) and summing the resulting equations for increasing m in a telescoping fash- ion, we arrive at the follo wing reduction. W e will use th e standard notation H ( a ) n := n X k =1 k − a for gener alize d harmonic sums . Theorem 2. F or i nte gers m ≥ 0 , Ls ( k ) n (2 mπ ) = k X j =0 ( − 1) k − j (2 π ) j k j ! H ( − j ) m Ls ( k − j ) n − j (2 π ) . Summarizing, we h a ve thus shown that the generalized log-sine integra ls at multiples of 2 π may alwa y s b e ev aluated in terms of integ rals at 2 π . In particular, Ls ( k ) n (2 mπ ) can alw a ys b e ev aluated in terms of zeta v al ues by the meth ods of Section 3.4 . 4.2 Reduct ion of arguments A general (real) argument σ can b e written uniquely as σ = 2 mπ ± σ 0 where m ≥ 0 is an integer and 0 ≤ σ 0 ≤ π . It then follo ws from ( 19 ) and Ls ( k ) n ( − θ ) = ( − 1) k +1 Ls ( k ) n ( θ ) that Ls ( k ) n ( σ ) equ als Ls ( k ) n (2 mπ ) ± k X j =0 ( ± 1) k − j (2 mπ ) j k j ! Ls ( k − j ) n − j ( σ 0 ) . (21) Since the ev aluatio n of log-sine integral s at multiples of 2 π w as explicitly treated in Section 4.1 this implies th at the ev aluatio n of log-sine integrals at general argumen ts σ re- duces to the case of argumen ts 0 ≤ σ ≤ π . 5. EV ALU A TIONS A T O THER V ALUE S In this section we fi rst d iscuss a metho d for ev aluating the generalized log-sine integra ls at arbitrary arguments in terms of Nielsen p olylogarithms at related arguments. The gist of our technique originates with F u c hs ([ 12 ], [ 20 , § 7.10]). Related ev aluations app ear in [ 8 ] for Ls 3 ( τ ) to Ls 6 ( τ ) as wel l as in [ 9 ] for Ls n ( τ ) and Ls (1) n ( τ ). W e then sp ecialize to ev aluations at π / 3 in Section 5.1 . The p olylogarithms arising in this case h a ve b een studied under the name of m ul tiple C l ausen and Glaisher values in [ 4 ]. In fact, the nex t result ( 22 ) with τ = π / 3 is a mo dified versi on of [ 4 , Lemma 3.2]. W e emplo y the notation n a 1 , . . . , a k ! := n ! a 1 ! · · · a k !( n − a 1 − . . . − a k )! for multinomial co efficien ts. Theorem 3. F or 0 ≤ τ ≤ 2 π , and nonne gative inte gers n , k such that n − k ≥ 2 , ζ ( n − k , { 1 } k ) − k X j =0 ( − iτ ) j j ! Li 2+ k − j, { 1 } n − k − 2 (e iτ ) = i k +1 ( − 1) n − 1 ( n − 1)! n − k − 1 X r =0 r X m =0 n − 1 k, m, r − m ! ×  i 2  r ( − π ) r − m Ls ( k + m ) n − ( r − m ) ( τ ) . (22) Pr oof. Starting with Li k, { 1 } n ( α ) − Li k, { 1 } n (1) = Z α 1 Li k − 1 , { 1 } n ( z ) z d z and integ rating by parts rep eatedly , w e obtain k − 2 X j =0 ( − 1) j j ! log j ( α ) Li k − j, { 1 } n ( α ) − Li k, { 1 } n (1) = ( − 1) k − 2 ( k − 2)! Z α 1 log k − 2 ( z ) Li { 1 } n +1 ( z ) z d z . (23) Letting α = e iτ and changing v ariables to z = e iθ , as well as using Li { 1 } n ( z ) = ( − log(1 − z )) n n ! , the righ t-hand side of ( 23 ) can b e rewritten as ( − 1) k − 2 ( k − 2)! i ( n + 1)! Z τ 0 ( iθ ) k − 2  − log  1 − e iθ  n +1 d θ . Since, for 0 ≤ θ ≤ 2 π and th e principal b ranc h of t he loga- rithm, log( 1 − e iθ ) = log     2 sin θ 2     + i 2 ( θ − π ) , (24) this last integral can n ow b e exp anded in t erms of general- ized log-sine in tegrals at τ . ζ ( k , { 1 } n ) − k − 2 X j =0 ( − iτ ) j j ! Li k − j, { 1 } n (e iτ ) = ( − i ) k − 1 ( k − 2)! ( − 1) n ( n + 1)! n +1 X r =0 r X m =0 n + 1 r ! r m !  i 2  r ( − π ) r − m Ls ( k + m − 2) n + k − ( r − m ) ( τ ) . (25) Applying the M ZV duali ty formula [ 3 ], we hav e ζ ( k , { 1 } n ) = ζ ( n + 2 , { 1 } k − 2 ) , and a c hange of v aria bles yields the claim. W e recall that the real and imaginary parts of the mul- tiple p olylogarithms are Clausen and Glaisher functions as defined in ( 4 ) and ( 5 ). Example 7. Apply ing ( 22 ) with n = 4 and k = 1 and solving for Ls (1) 4 ( τ ) yields Ls (1) 4 ( τ ) = 2 ζ (3 , 1) − 2 Gl 3 , 1 ( τ ) − 2 τ Gl 2 , 1 ( τ ) + 1 4 Ls (3) 4 ( τ ) − 1 2 π Ls (2) 3 ( τ ) + 1 4 π 2 Ls (1) 2 ( τ ) = 1 180 π 4 − 2 Gl 3 , 1 ( τ ) − 2 τ Gl 2 , 1 ( τ ) − 1 16 τ 4 + 1 6 π τ 3 − 1 8 π 2 τ 2 . F or the last eq ualit y we used the t riv ial ev aluation Ls ( n − 1) n ( τ ) = − τ n n . (26) It app ears that b oth Gl 2 , 1 ( τ ) and Gl 3 , 1 ( τ ) are not reducible for τ = π / 2 or τ = 2 π / 3. Here, reducible means express- ible in terms of multi zeta v alues and Glaisher functions of the same argument and low er wei ght. In the case τ = π / 3 such reductions are p ossible. This is discussed in Example 9 and illustrates how muc h less simple v alues at 2 π/ 3 are than those at π / 3. W e remark, how ev er, that Gl 2 , 1 (2 π/ 3 ) is re- ducible to one-dimensional p olylogarithmic terms [ 6 ]. In [ 1 ] explicit reductions for all weigh t four or less p olylogarithms are giv en. ✸ R emark 3. Lewin [ 20 , 7.4.3] uses the special case k = n − 2 of ( 22 ) to deduce a few small integer ev aluatio ns of the log-sine integrals Ls ( n − 2) n ( π / 3) in t erms of classical Clausen functions. ✸ In general, w e can use ( 22 ) recursiv ely to express the log-sine v alues Ls ( k ) n ( τ ) in terms of multiple Clausen and Glaisher functions at τ . Example 8. ( 22 ) with n = 5 and k = 1 p roduces Ls (1) 5 ( τ ) = − 6 ζ (4 , 1) + 6 Cl 3 , 1 , 1 ( τ ) + 6 τ Cl 2 , 1 , 1 ( τ ) + 3 4 Ls (3) 5 ( τ ) − 3 2 π Ls (2) 4 ( τ ) + 3 4 π 2 Ls (1) 3 ( τ ) . Applying ( 22 ) three more times to rewrite the remaining log- sine integral s pro duces an ev aluation of Ls (1) 5 ( τ ) in terms of multi zeta v alues and Clausen functions at τ . ✸ 5.1 Log-sine integrals at π / 3 W e no w apply the general results obtained in Section 5 to the ev aluation of log-sine integrals at τ = π / 3. Accord- ingly , w e en counter multiple p olylogarithms at the basic 6- th ro ot of unity ω := exp( iπ / 3). Their real and imaginary parts satisfy v ario us relations and reductions, studied in [ 4 ], whic h allo w us to further treat the resulting ev aluations. In general, th ese polylogarithms are more tractable th an those at other v al ues b ecause ω = ω 2 . Example 9. (V alues at π 3 ) Con tin uing Example 7 w e have − Ls (1) 4  π 3  = 2 Gl 3 , 1  π 3  + 2 3 π Gl 2 , 1  π 3  + 19 6480 π 4 . Using known reductions from [ 4 ] we get: Gl 2 , 1  π 3  = 1 324 π 3 , Gl 3 , 1  π 3  = − 23 19440 π 4 , (27) and so arrive at − Ls (1) 4  π 3  = 17 6480 π 4 . (28) Lewin exp licitly mentions ( 28 ) in the preface to [ 20 ] b ecause of its “queer” nature whic h he compares to some of Landen ’s curious 18th century form ulas. ✸ Man y m ore reduction b esides ( 27 ) are k no wn. In par- ticular, the one-dimensional Glaisher and Clausen functions reduce as follo ws [ 20 ]: Gl n (2 π x ) = 2 n − 1 ( − 1) 1+ ⌊ n/ 2 ⌋ n ! B n ( x ) π n , Cl 2 n +1  π 3  = 1 2 (1 − 2 − 2 n )(1 − 3 − 2 n ) ζ (2 n + 1) . (29) Here, B n denotes th e n -th Bernoul li p olynomial . F u rther reductions can b e derived for instance from the dualit y result [ 4 , Theorem 4.4]. F or low d imensions, we hav e built th ese reductions into our program, see Section 6 . Example 10. (V alues of Ls n ( π / 3)) The log-sine integrals at π / 3 are ev aluated by our program as follo ws: Ls 2  π 3  = Cl 2  π 3  − Ls 3  π 3  = 7 108 π 3 Ls 4  π 3  = 1 2 π ζ (3) + 9 2 Cl 4  π 3  − Ls 5  π 3  = 1543 19440 π 5 − 6 Gl 4 , 1  π 3  Ls 6  π 3  = 15 2 π ζ (5) + 35 36 π 3 ζ (3) + 135 2 Cl 6  π 3  − Ls 7  π 3  = 74369 326592 π 7 + 15 2 π ζ (3) 2 − 135 Gl 6 , 1  π 3  As follo ws from the results of Section 5 each integ ral is a multiv ariable rational polynomial in π as well as Cl , Gl, and zeta v alues. These ev aluations confirm th ose giv en in [ 9 , A pp endix A ] for Ls 3  π 3  , Ls 4  π 3  , an d Ls 6  π 3  . Less explicitely , the ev aluations of Ls 5  π 3  and Ls 7  π 3  can b e reco vere d from similar results in [ 15 , 9 ] (whic h in part w ere obtained using PSLQ; we refer to Section 6 for ho w our analysis relies on PSLQ). The first presumed-irreducible val ue that occu rs is Gl 4 , 1  π 3  = ∞ X n =1 P n − 1 k =1 1 k n 4 sin  nπ 3  = 3341 1632960 π 5 − 1 π ζ (3) 2 − 3 4 π ∞ X n =1 1  2 n n  n 6 . (30) The fi nal ev aluation is describ ed in [ 4 ]. Ext en siv e compu- tation suggests it is not expressible as a sum of pro ducts of one dimensional Glaisher and zeta v alues. Indeed, con- jectures are made in [ 4 , § 5] for the num ber of irreducibles at eac h depth . Related dimensional conjectures for polylogs are discussed in [ 26 ]. ✸ 5.2 Log-sine integrals at imaginary values The approach of Section 5 may b e extended to ev aluate log-sine integral s at imaginary arguments. In more usual terminology , t h ese are l o g-sinh inte gr als Lsh ( k ) n ( σ ) := − Z σ 0 θ k log n − 1 − k     2 sinh θ 2     d θ (31) whic h are related to log-sine integ rals by Lsh ( k ) n ( σ ) = ( − i ) k +1 Ls ( k ) n ( iσ ) . W e may d erive a result along the lines of Theorem 3 by observing th at equation ( 24 ) is replaced, when θ = it for t > 0, by the simpler log(1 − e − t ) = log     2 sinh t 2     − t 2 . (32) This leads t o: Theorem 4. F or t > 0 , and nonne gat ive inte gers n , k such that n − k ≥ 2 , ζ ( n − k , { 1 } k ) − k X j =0 t j j ! Li 2+ k − j, { 1 } n − k − 2 (e − t ) = ( − 1) n + k ( n − 1)! n − k − 1 X r =0 n − 1 k, r !  − 1 2  r Lsh ( k + r ) n ( t ) . (33) Example 11. Let ρ := (1 + √ 5) / 2 b e the golden mean. Then, by applying Theorem 4 with n = 3 and k = 1, Lsh (1) 3 (2 log ρ ) = ζ (3) − 4 3 log 3 ρ − Li 3 ( ρ − 2 ) − 2 Li 2 ( ρ − 2 ) log ρ . This may b e further reduced, using Li 2 ( ρ − 2 ) = π 2 15 − log 2 ρ and Li 3 ( ρ − 2 ) = 4 5 ζ (3) − 2 15 π 2 log ρ + 2 3 log 3 ρ , to yield the w ell-know n Lsh (1) 3 (2 log ρ ) = 1 5 ζ (3) . The interest in this kind of ev al uation stems from the fact that log-sinh integrals at 2 log ρ express v alues of alternating inv erse binomial sums (the fact that log-sine in tegrals at π / 3 giv e inv erse binomial sums is illustrated by Example 10 and ( 30 )). I n t his case, Lsh (1) 3 (2 log ρ ) = 1 2 ∞ X n =1 ( − 1) n − 1  2 n n  n 3 . More on this relation and generalizations can b e found in eac h of [ 22 , 16 , 4 , 2 ]. ✸ 6. REDUCING POL YLOGARITHMS The techniques describ ed in Sections 3.3 and 5 for ev alu- ating log-sine integrals in terms of multiple p olylogarithms usually produce expressions that can be considerably re- duced as is illustrated in Examples 3 , 4 , and 9 . R elations b et w een p olylogarithms hav e b een the sub ject of many stud- ies [ 3 , 2 ] with a sp ecial focu s on (alternating) multiple zeta v alues [ 17 , 13 , 26 ] and, to a lesser extent, Clausen v alues [ 4 ]. There is a certain d eal of choice in how to combine the v ario us techniques th at we present in order to eva luate log- sine integral s at certain v alues. The next example shows how this can b e exp loited to d eriv e relations among the va rious p olylogari thms invol ved. Example 12. F or n = 5 and k = 2, sp ecializing ( 19 ) to σ = π and m = 1 yields Ls (2) 5 (2 π ) = 2 Ls (2) 5 ( π ) − 4 π Ls (1) 4 ( π ) + 4 π 2 Ls 3 ( π ) . By Example 5 w e know th at this ev aluates as − 13 / 45 π 5 . On the other hand, we ma y use th e tec hnique of Section 3.3 to reduce the log-sine integ rals at π . This leads to − 8 π Li 3 , 1 (1) + 12 π Li 4 (1) − 2 5 π 5 = − 13 45 π 5 . In s implified terms, w e hav e d eriv ed th e famous identit y ζ (3 , 1) = π 4 360 . Similarly , the case n = 6 and k = 2 leads to ζ (3 , 1 , 1) = 3 2 ζ (4 , 1) + 1 12 π 2 ζ (3) − ζ (5) which further reduces to 2 ζ (5) − π 2 6 ζ (3). A s a final example, the case n = 7 and k = 4 p roduces ζ (5 , 1) = π 6 1260 − 1 2 ζ (3) 2 . ✸ F or the purp ose of an implementation, we h a v e built many reductions of multiple p olylogarithms into our program. Be- sides some general ru les, su c h as ( 29 ), the program contains a table of reductions at lo w weigh t for p olylogarithms at the v alues 1 and − 1, as well as Clausen and Glaisher functions at the v alues π / 2, π / 2, and 2 π / 3. T hese corresp ond to th e p olylogari thms th at o ccur in the ev aluation of t he log-sine integ rals at the special val ues π / 3, π / 2, 2 π / 3, π whic h are of particular imp ortance for applications as mentioned in the introduction. This table of reductions h as b een compiled using the integer relation finding algorithm PSLQ [ 2 ]. It s use is thus of heuristic nature (as opp osed to the rest of the program which is working sy m b olical ly from the analyt ic results in this pap er) and is t herefore made optional. 7. THE PR OGRAM 7.1 Basic usage As promised, we implemented 1 the presented results for ev aluating log-sine integrals for use in the computer algebra systems Mathematic a an d SAGE. The basic usage is very simple and illustrated in the next example for Mathematic a 2 . Example 13. Consider the log-sine integ ral Ls (2) 5 (2 π ). The follo wi ng self-explanatory code ev al uates it in terms of p oly- logarithms: L s T o L i [ L s [ 5 , 2 , 2 P i ] ] This pro duces the output − 13 / 45 π 5 as in Example 5 . As a second example, - L s T o L i [ L s [ 5 , 0 , P i / 3 ] ] results in t he output 1 5 4 3 / 1 9 4 4 0 * P i ^ 5 - 6 * G l [ { 4 , 1 } , P i / 3 ] whic h agrees with the ev aluation in Example 10 . Finally , L s T o L i [ L s [ 5 , 1 , P i ] ] prod uces 6 * L i [ { 3 , 1 , 1 } , - 1 ] + ( P i ^ 2 * Z e t a [ 3 ] ) / 4 - ( 1 0 5 * Z e t a [ 5 ] ) / 3 2 as in Example 3 . ✸ Example 14. Computing L s T o L i [ L s [ 6 , 3 , P i / 3 ] - 2 * L s [ 6 , 1 , P i / 3 ] ] yields the va lue 313 20412 0 π 6 and thus automatically prov es a result of Zuck er [ 27 ]. A family of relations b etw een log-sine integ rals at π / 3 generalizing t he above h as been established in [ 22 ]. ✸ 7.2 Implementation The conv ersio n from log-sine integ rals to polylogarithmic v alues demonstrated in Example 13 roughly p ro ceeds as fol- lo ws: • First, the ev aluation of Ls ( k ) n ( σ ) is reduced to the cases of 0 ≤ σ ≤ π and σ = 2 mπ as described in S ection 4.2 . • The cases σ = 2 mπ are t reated as in Section 3.4 and result in m ultiple zeta val ues. • The other cases σ result in p olylogari thmic v alues at e iσ and are obtained using the results of S ections 3.3 and 5 . • Finally , especially in th e physically relev ant cases, v ar- ious reductions of the resulting p olylogarithms are p er- formed as outlined in Section 6 . 1 The pack ages are freely a v ailable for download from http://arm instraub.c om/pub/log - sine- integrals 2 The interfac e in the case of SAGE is similar but may change sligh tly , especially as we hop e to integrate our pack age into the core of SAGE. 7.3 Numerical usage The program is also u seful for numerical computations provided that it is coupled with efficient metho ds for ev al- uating p olylogarithms t o high p recision. It complements for instance th e C++ library lsjk “for arbitrary- precision numeric ev aluation of the generalize d log-sine functions” de- scribed in [ 15 ]. Example 15. W e ev aluate Ls (2) 5  2 π 3  = 4 Gl 4 , 1  2 π 3  − 8 3 π Gl 3 , 1  2 π 3  − 8 9 π 2 Gl 2 , 1  2 π 3  − 8 1215 π 5 . Using sp ecialize d co de 3 such as [ 25 ], t h e right-hand side is readily eval uated to, for instance, tw o thousand digit pre- cision in ab out a minute. The first 1024 digits of the re- sult match the ev aluation given in [ 15 ]. Ho w eve r, due to its implementati on lsjk currently is restricted to log-sine functions Ls ( k ) n ( θ ) with k ≤ 9. ✸ Acknowledgements. W e are grateful to An drei Davydychev and Mikh ail Kal- myk o v for several v aluable comments on an earlier versi on of this pap er and for p ointing us to relev an t publications. W e also thank the review ers for their th orough reading and helpful suggestions. 8. REFERENCES [1] D. Borw ein, J. M. Borw ein, A. 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