Explicit formulae

It is convenient to introduce some special notation: we will write 𝐴_𝑛,𝑟(𝜏) ∶=𝐴(𝑛,0,…,0

⏟⏟⏟

𝑟−1

;𝜏) =

∫

1 0

(2𝜋𝑖𝑡)^𝑟−1

(𝑟− 1)! 𝑓_𝑛(𝑡, 𝜏)𝑑𝑡, and

(𝑋, 𝑌;𝜏) ∶=∑

𝑛≥0 𝑟≥1

𝐴_𝑛,𝑟(𝜏)

(2𝜋𝑖)^𝑟−1𝑋^𝑛−1𝑌^𝑟−1 = ∑

𝑟≥1∫

1 0

(𝑡𝑌)^𝑟−1 (𝑟− 1)!𝐹

( 𝑡, 𝑋

2𝜋𝑖;𝜏 )

𝑑𝑡

= ∫

1 0

𝑒^𝑡𝑌𝐹 (

𝑡, 𝑋 2𝜋𝑖;𝜏

) 𝑑𝑡,

where we mention, once for all, that by∫₀¹we mean theregularized integraldescribed in the definition of elliptic MZVs.

First of all, note that one can reduce all depth one elliptic MZVs to the𝐴_𝑛,𝑟’s:

𝐴(0,…,0

⏟⏟⏟

𝑠

, 𝑛,0,…,0

⏟⏟⏟

𝑟

;𝜏) =

= (2𝜋𝑖)^𝑟+𝑠

∫

1 0

𝑡^𝑟(1 −𝑡)^𝑠

𝑟!𝑠! 𝑓_𝑛(𝑡, 𝜏) =

∑𝑠 𝑖=0

(2𝜋𝑖)^𝑠−𝑖(−1)^𝑖 (𝑠−𝑖)!

(𝑟+𝑖 𝑟

)

𝐴_{𝑛,𝑟+𝑖+1}(𝜏). (5.87) Let us now see what is the analogue of Proposition5.3.1for the depth 1 case.

Lemma 5.5.1.

𝜕

𝜕𝜏(𝑋, 𝑌;𝜏) = (1 −𝑒^𝑌)(𝑋, 𝜏) −𝑌 𝜕

𝜕𝑋(𝑋, 𝑌;𝜏) (5.88) Proof. Using the mixed heat equation, the substitution𝑋̂ = 𝑋∕2𝜋𝑖and Lemma 5.3.1we get¹⁷

𝜕

𝜕𝜏 ∫

1 0

𝑒^𝑡𝑌𝐹 (

𝑡, 𝑋 2𝜋𝑖;𝜏

)

𝑑𝑡= 1 2𝜋𝑖∫

1 0

𝑒^𝑡𝑌 𝜕

𝜕𝑡𝜕 ̂𝑋

𝐹(𝑡, ̂𝑋;𝜏)𝑑𝑡

= 1 2𝜋𝑖

( 𝑒^𝑌 𝜕

𝜕 ̂𝑋

𝐹(1, ̂𝑋;𝜏) − 𝜕

𝜕 ̂𝑋

𝐹(0, ̂𝑋;𝜏) )

− 𝑌 2𝜋𝑖∫

1 0

𝑒^𝑡𝑌 𝜕

𝜕 ̂𝑋

𝐹(𝑡, ̂𝑋;𝜏)𝑑𝑡

= (1 −𝑒^𝑌)(𝑋, 𝜏) −𝑌 𝜕

𝜕𝑋(𝑋, 𝑌;𝜏). (5.89)

□

We want to make more precise in this setting the remark that elliptic MZVs can be written as iterated integrals of Eisenstein series. As always, we write𝐴_𝑛,𝑟(𝜏) = 𝐴^∞_𝑛,𝑟+𝐴⁰_𝑛,𝑟(𝜏), as well as(𝑋, 𝑌;𝜏) =^∞(𝑋, 𝑌) +⁰(𝑋, 𝑌;𝜏)at the level of generating

17As in the proof of Enriquez’s Proposition5.3.1, we omit the proof that the basic properties of iterated integrals are still valid for the regularized ones. All details can be found in [44].

functions. Note that, by (5.37), we have that for all𝑛≠1 𝐴^∞_𝑛,𝑟= (2𝜋𝑖)^𝑟B_𝑛

𝑟!𝑛! , (5.90)

and that, by Corollary5.2.1and (5.22),

𝐴^∞_1,𝑟= (2𝜋𝑖)^𝑟−1( 𝑖𝜋 2(𝑟− 1)! −

⌊^𝑟₂⌋−1

∑

𝑘=1

𝜁(2𝑘+ 1) (𝑟− 2𝑘− 1)!(2𝜋𝑖)^2𝑘

)

. (5.91)

We will denote the generating series of the exponentially small part𝔾⁰_𝑘 of the Eisen-stein series𝔾_𝑘 by

⁰(𝑋, 𝜏) ∶=∑

𝑛≥1

𝑛𝔾⁰_𝑛+1𝑋^𝑛−1. (5.92)

Moreover, we denote¹⁸

Γ^𝐿_𝑛,𝑘(𝜏) ∶= Γ(0,…,0

⏟⏟⏟

𝑘−1

, 𝑛;𝜏) =

∫[𝜏,⃗1∞]

𝔾_𝑛𝔾₀⋯𝔾₀

⏟⏞⏟⏞⏟

𝑘−1

and

Γ^𝑅_𝑛,𝑘(𝜏) ∶= Γ(𝑛,0,…,0

⏟⏟⏟

𝑘−1

;𝜏) =

∫[𝜏,⃗1∞]

𝔾₀⋯𝔾₀

⏟⏞⏟⏞⏟

𝑘−1

𝔾_𝑛

Again, we write

Γ^𝐿_𝑛,𝑘(𝜏) = Γ^𝐿,∞_𝑛,𝑘 (𝜏) + Γ^𝐿,0_𝑛,𝑘(𝜏), Γ^𝑅_𝑛,𝑘(𝜏) = Γ^𝑅,∞_𝑛,𝑘 (𝜏) + Γ^𝑅,0_𝑛,𝑘(𝜏).

It is obvious that

Γ^𝐿,0_𝑛,𝑘(𝜏) =

∫_{[𝜏,𝑖∞]}𝔾⁰_𝑛𝔾₀⋯𝔾₀

⏟⏞⏟⏞⏟

𝑘−1

(5.93)

and that

Γ^𝐿,∞_𝑛,𝑘 (𝜏) = B_𝑛(2𝜋𝑖𝜏)^𝑘

𝑛!𝑘! . (5.94)

Moreover, one can easily check that Γ^𝑅,0_𝑛,𝑘(𝜏) = (2𝜋𝑖)^𝑘−1

(𝑘− 1)! ∫

𝑖∞

𝜏

𝑧^𝑘−1𝔾⁰_𝑛(𝑧)𝑑𝑧 (5.95) and that

Γ^𝑅,∞_𝑛,𝑘 (𝜏) = Γ^𝐿,∞_𝑛,𝑘 (𝜏) = B_𝑛(2𝜋𝑖𝜏)^𝑘

𝑛!𝑘! . (5.96)

It is important to keep in mind thatΓ^𝐿_𝑛,𝑘(𝜏)andΓ^𝑅_𝑛,𝑘(𝜏)vanish identically whenever𝑛 is odd. Moreover, we want to stress the fact that one can compute the𝑞-expansion of anyΓ⁰(𝑛₁,…, 𝑛_𝑟;𝜏), and this is the simplest way to get the𝑞-expansion of A-elliptic MZVs, as already noticed in [16]. In depth one the𝑞-expansion (5.85) just reads

Γ^𝐿,0_𝑛,𝑘(𝜏) = − 2 (𝑛− 1)!

∑

𝑚,𝑝≥1

𝑚^{𝑛−𝑘−1}

𝑝^𝑘 𝑞^𝑚𝑝. (5.97)

18As we have already done in the previous section, here by∫ 𝔾_𝑘we mean∫ 𝔾_𝑘(𝑧)𝑑𝑧.

Let us now introduce the generating functions

(𝑋, 𝑌;𝜏) ∶= ∑

𝑛,𝑘≥1

(−1)^𝑛−1 (2𝜋𝑖)^𝑘−1

(𝑛+𝑘− 1)!

(𝑛− 1)! Γ^𝐿_{𝑛+𝑘,𝑘}(𝜏)𝑋^𝑛−1𝑌^𝑘−1, (5.98)

(𝑋, 𝑌;𝜏) ∶= ∑

𝑛,𝑘≥1

1 (2𝜋𝑖)^𝑘−1

(𝑛+𝑘− 1)!

(𝑛− 1)! Γ^𝑅_{𝑛+𝑘,𝑘}(𝜏)𝑋^𝑛−1𝑌^𝑘−1. (5.99) As before, we write

⁰(𝑋, 𝑌;𝜏) = ∑

𝑛,𝑘≥1

(−1)^𝑛−1 (2𝜋𝑖)^𝑘−1

(𝑛+𝑘− 1)!

(𝑛− 1)! Γ^𝐿,0_{𝑛+𝑘,𝑘}(𝜏)𝑋^𝑛−1𝑌^𝑘−1 and

⁰(𝑋, 𝑌;𝜏) = ∑

𝑛,𝑘≥1

1 (2𝜋𝑖)^𝑘−1

(𝑛+𝑘− 1)!

(𝑛− 1)! Γ^𝑅,0_{𝑛+𝑘,𝑘}(𝜏)𝑋^𝑛−1𝑌^𝑘−1.

The main consequence of the lemma above (this consequence, stated in a different way, is already contained in [66]) is the following:

Proposition 5.5.1.

⁰(𝑋, 𝑌;𝜏) = (𝑒^𝑌 − 1)⁰(𝑋, 𝑌;𝜏) (5.100) Proof.Iterating the statement of the proposition, and noting that, for every𝑘≥1,

∫_[𝜏,⃗1_∞]

𝐴^∞_𝑛,𝑟𝑑𝑧₁𝑑𝑧₂⋯𝑑𝑧_𝑘 =𝑂(𝑞), we deduce that

⁰(𝑋, 𝑌;𝜏) = (𝑒^𝑌 − 1)∑

𝑘≥1

𝑌^𝑘−1 (2𝜋𝑖)^𝑘−1

𝜕^𝑘−1

𝜕𝑋^𝑘−1∫[𝜏,𝑖∞]⁰(𝑋, 𝑧₁)𝑑𝑧₁𝔾₀𝑑𝑧₂⋯𝔾₀𝑑𝑧_𝑘. (5.101) Expanding the generating series⁰, we get

⁰(𝑋, 𝑌;𝜏) = (𝑒^𝑌 − 1) ∑

𝑘,𝑚≥1

(−1)^𝑘−1𝑚!

(𝑚−𝑘)!(2𝜋𝑖)^𝑘−1Γ^𝐿,0_𝑚+1,𝑘(𝜏)𝑋^𝑚−𝑘𝑌^𝑘−1

= (𝑒^𝑌 − 1) ∑

𝑘,𝑛≥1

(−1)^𝑛−1(𝑛+𝑘− 1)!

(𝑛− 1)!(2𝜋𝑖)^𝑘−1 Γ^𝐿,0_{𝑛+𝑘,𝑘}(𝜏)𝑋^𝑛−1𝑌^𝑘−1,

where to get(−1)^𝑛−1 in the second equality we have used that𝑛+𝑘must be even.

This concludes the proof.

□

Corollary 5.5.1. For any𝑛≥1and any𝑟≥1we have 𝐴_𝑛,𝑟(𝜏) =𝐴^∞_𝑛,𝑟+(−1)^𝑛−1

(𝑛− 1)!

∑𝑟−1 𝑗=1

(2𝜋𝑖)^𝑟−𝑗(𝑛+𝑗− 1)!

(𝑟−𝑗)! Γ^𝐿,0_{𝑛+𝑗,𝑗}(𝜏). (5.102) Proof. It is a straightforward comparison term by term of the left hand side of the proposition with the right hand side.

□

Remark 5.5.2. This corollary, together with (5.97), the formulae for the constant term 𝐴^∞_𝑛,𝑟 and formula (5.87), lead to completely explicit formulae for the 𝑞-expansion of

all A-elliptic MZVs of depth one, which allow to approximate them numerically to high precision¹⁹. We have used this to check numerically all results given in the rest of this section.

Remark 5.5.3. By formula (5.102) we deduce that all A-elliptic MZVs of the form 𝐴_1,𝑟(𝜏)includeΓ(2, 𝜏) = −2 log(𝜂(𝜏))as part of their Fourier expansion, and therefore all𝐵_1,𝑟(𝜏) ∶= 𝐵(1,0,…,0

⏟⏟⏟

𝑟−1

;𝜏) will include log(𝜏) in the asymptotic expansion. Note that for the modified

𝐴̂_1,𝑟(𝜏) ∶=𝐴_1,𝑟(𝜏) −(2𝜋𝑖)^𝑟−2

(𝑟− 1)!𝐴_1,2(𝜏) (5.103) the termΓ(2, 𝜏)disappears.

Another consequence of the proposition above is that we can easily invert the rôle of A-elliptic MZVs and iterated integrals of Eisenstein series:

Corollary 5.5.2. For any𝑛≥1and any1≤𝑘≤𝑛− 1we have Γ^𝐿,0_𝑛,𝑘(𝜏) = (−1)^{𝑛−𝑘−1}(𝑛−𝑘− 1)!

(𝑛− 1)!

𝑘+1∑

𝑖=2

B_{𝑘+1−𝑖}(2𝜋𝑖)^{𝑛−𝑘−𝑖}

(𝑘+ 1 −𝑖)! 𝐴⁰_{𝑛−𝑘,𝑖}(𝜏). (5.104) Proof. Note that𝑒^𝑌 − 1 = 𝑌 +𝑌²∕2 + …, and this corresponds to the fact that length one elliptic MZVs are constant. This means that, if we want to invert equation (5.100), we need to rescale both sides, i.e. to multiply both sides by𝑌. This yields

𝑌

𝑒^𝑌 − 1⁰(𝑋, 𝑌;𝜏) =⁰(𝑋, 𝑌;𝜏), (5.105) and comparing term by term this equation we are done.

□

Now we want to deduce explicit formulae for B-elliptic MZVs of depth one. To do this, we need to know the behaviour ofΓ^𝐿_𝑛,𝑘(𝜏)under𝜏 ↦−1∕𝜏. Let us introduce another piece of notation. For any operator Λ acting on a space of formal power series, we setΛ^(𝑘) ∶= Λ◦ ⋯ ◦Λ

⏟⏞⏟⏞⏟

𝑘

, and exp(Λ) ∶= ∑

𝑘≥0Λ^(𝑘)∕𝑘!. Then we have the fol-lowing two lemmas.

Lemma 5.5.4.

⁰(𝑋, 𝑌;𝜏) =⁰(𝑋+𝜏𝑌 , 𝑌;𝜏) (5.106) Proof.By equation (5.95), we can write

⁰(𝑋, 𝑌;𝜏) =∑

𝑘≥1

𝑌^𝑘−1 (𝑘− 1)!

𝜕^𝑘−1

𝜕𝑋^𝑘−1∫

𝑖∞

𝜏

𝑧^𝑘−1⁰(𝑋, 𝑧)𝑑𝑧 (5.107)

19One can easily get500digits in less than a second with PARI GP.

Since𝑑∫_{[𝜏,𝑖∞]}𝔾⁰_𝑘(𝑧)𝑑𝑧= −𝔾⁰_𝑘(𝜏), repeatedly integrating by parts gives

∑

𝑘≥1

𝑌^𝑘−1 (𝑘− 1)!

𝜕^𝑘−1

𝜕𝑋^𝑘−1 ∫

𝑖∞

𝜏

𝑧^𝑘−1⁰(𝑋, 𝑧)𝑑𝑧=

= exp( 𝜏𝑌 𝜕

𝜕𝑋 ) ∑

𝑘≥1

𝑌^𝑘−1 (2𝜋𝑖)^𝑘−1

𝜕^𝑘−1

𝜕𝑋^𝑘−1 ∫[𝜏,𝑖∞]⁰(𝑋, 𝑧₁)𝑑𝑧₁𝔾₀𝑑𝑧₂⋯𝔾₀𝑑𝑧_𝑘, (5.108) and this, as we have already seen in the proof of Proposition5.5.1, leads to the iden-tity

⁰(𝑋, 𝑌;𝜏) = exp( 𝜏𝑌 𝜕

𝜕𝑋 )

⁰(𝑋, 𝑌;𝜏).

By Taylor’s theoremexp( 𝑎^d

dx

)𝑓(𝑥) =𝑓(𝑥+𝑎). This concludes the proof.

□

By an abuse of notation, we denote by ⁰(𝑋, 𝑌; −1∕𝜏) the exponentially small (with respect to𝜏 → 𝑖∞) part of the function⁰(𝑋, 𝑌; −1∕𝜏).²⁰ This will not create any confusion in what follows.

Lemma 5.5.5.

⁰(𝑋, 𝑌; −1∕𝜏) =⁰(𝑌 ,−𝑋;𝜏) (5.109) Proof.We have already seen that we can write

⁰(𝑋, 𝑌;𝜏) =∑

𝑘≥1

𝑌^𝑘−1 (𝑘− 1)!

𝜕^𝑘−1

𝜕𝑋^𝑘−1∫

𝑖∞

𝜏

𝑧^𝑘−1⁰(𝑋, 𝑧)𝑑𝑧.

Therefore, using the transformation properties of, we find that the exponentially small part of⁰(𝑋, 𝑌; −1∕𝜏)is equal to

∑

𝑘≥1

(−1)^𝑘−1𝑌^𝑘−1 (𝑘− 1)!

𝜕^𝑘−1

𝜕𝑋^𝑘−1∫

𝑖∞

𝜏

𝑧^1−𝑘⁰(𝑧𝑋, 𝑧)𝑑𝑧.

Expanding this expression with respect to𝑋, and using that𝑛+𝑘must be even, we

find ∑

𝑘,𝑛≥1

(−1)^𝑛−1(𝑛+𝑘− 1)!

(𝑛− 1)!(𝑘− 1)! ∫

𝑖∞

𝜏

𝑧^𝑛−1𝔾⁰_𝑛+𝑘(𝑧)𝑑𝑧 𝑌^𝑘−1𝑋^𝑛−1. Comparing this expression with the definition ofconcludes the proof.

□

Comparing the coefficients of the identity in Lemma5.5.4gives Corollary 5.5.3. For each𝑛, 𝑗 ≥1we have

Γ^𝑅,0_{𝑛+𝑗,𝑗}(𝜏) =

∑𝑗−1 𝑖=0

(−1)^{𝑛+𝑖−1}(2𝜋𝑖𝜏)^𝑖

𝑖! Γ^𝐿,0_{𝑛+𝑗,𝑗−𝑖}(𝜏), (5.110)

Γ^𝐿,0_{𝑛+𝑗,𝑗}(𝜏) =

∑𝑗−1 𝑖=0

(−1)^{𝑛+𝑖−1}(2𝜋𝑖𝜏)^𝑖

𝑖! Γ^𝑅,0_{𝑛+𝑗,𝑗−𝑖}(𝜏). (5.111)

Moreover, Lemma5.5.5implies

20In general the latter yields some non-exponentially small term, as we will see later.

Corollary 5.5.4. For𝑛, 𝑗 ≥1,

Γ^𝑅,0_{𝑛+𝑗,𝑗}(−1∕𝜏) = (−1)^𝑛−1(2𝜋𝑖)^𝑗−𝑛(𝑛− 1)!

(𝑗− 1)! Γ^𝑅,0_{𝑛+𝑗,𝑛}(𝜏) (5.112) These two facts together lead to

Corollary 5.5.5. Let𝑛≥1and1≤𝑗 ≤𝑛− 1. Then Γ^𝑅,0_𝑛,𝑗(−1∕𝜏) = (2𝜋𝑖)^𝑗−1(𝑛− 1 −𝑗)!

(𝑗− 1)!

∑𝑛−𝑗 𝑖=1

(−1)^𝑖+𝑗𝜏^{𝑛−𝑗−𝑖}

(𝑛−𝑗−𝑖)!(2𝜋𝑖)^𝑖−1Γ^𝐿,0_𝑛,𝑖 (𝜏) (5.113) and

Γ^𝐿,0_𝑛,𝑗(−1∕𝜏) = (2𝜋𝑖)^𝑗−1

∑𝑗−1 𝑖=0

(𝑛−𝑗+𝑖− 1)!

𝑖!(𝑗−𝑖− 1)!

𝑛−𝑗+𝑖∑

𝑘=1

(−1)^𝑘−1𝜏^{𝑛−𝑗−𝑘}

(𝑛−𝑗+𝑖−𝑘)!(2𝜋𝑖)^𝑘−1Γ^𝐿,0_𝑛,𝑘(𝜏) (5.114) We are now ready to put all the pieces together. Let us consider

𝐵_𝑛,𝑟(𝜏) =𝐵(𝑛,0,…,0

⏟⏟⏟

𝑟−1

;𝜏).

We write

𝐵_𝑛,𝑟(𝜏) =𝐵_𝑛,𝑟^∞(𝜏) +𝐵⁰_𝑛,𝑟(𝜏),

meaning as always that𝐵_𝑛,𝑟^∞(𝜏)denotes the first Laurent polynomials in the asymp-totic expansion of𝐵_𝑛,𝑟(𝜏)(plus a logarithmic term, in case𝑛= 1, cf. Remark5.5.3) and 𝐵_𝑛,𝑟⁰ (𝜏)denotes the exponentially small part. Let us consider the generating function

⁰(𝑋, 𝑌;𝜏) ∶= ∑

𝑛,𝑟≥1

𝐵⁰_𝑛,𝑟(𝜏)

(2𝜋𝑖)^𝑟−1𝑋^𝑛−1𝑌^𝑟−1. Then we have

Theorem 5.5.6.

⁰(𝑋, 𝑌;𝜏) = (𝑒^𝑌 − 1)⁰(

−𝜏𝑋,−𝑋− 𝑌 𝜏;𝜏

)

. (5.115)

Proof.By Proposition5.5.1and Lemma5.5.4we have

⁰(𝑋, 𝑌; −1∕𝜏) = (𝑒^𝑌 − 1)⁰(𝑋, 𝑌; −1∕𝜏) = (𝑒^𝑌 − 1)⁰( 𝑋+ 𝑌

𝜏, 𝑌; −1∕𝜏 )

, where in this case⁰(𝑋, 𝑌; −1∕𝜏)really means that we consider⁰(𝑋, 𝑌;𝜏)and then we invert𝜏. Using Lemma5.5.5and again Lemma5.5.4we conclude that

⁰(𝑋, 𝑌;𝜏) = (𝑒^𝑌 − 1)⁰(

𝑌 ,−𝑋−𝑌 𝜏;𝜏

)

= (𝑒^𝑌 − 1)⁰(

−𝜏𝑋,−𝑋−𝑌 𝜏;𝜏

) , where in this case⁰(𝑋, 𝑌; −1∕𝜏) was meant to be the exponentially small part of

⁰(𝑋, 𝑌; −1∕𝜏).

□

The consequence of this theorem is that we are now able to express B-elliptic MZVs of depth one in terms of iterated integrals of Eisenstein series, and therefore in terms of A-elliptic MZVs:

Corollary 5.5.6. For𝑛, 𝑟≥1we have 𝐵_𝑛,𝑟(𝜏) =𝐵_𝑛,𝑟^∞(𝜏)+

(2𝜋𝑖)^𝑟−1 (𝑛− 1)!

∑𝑟−1 𝑗=1

(𝑛+𝑗− 1)!

(𝑟−𝑗)!

∑𝑗−1 𝑖=0

(−1)^{𝑗−𝑖−1}(𝑛+𝑖− 1)!

𝑖!(𝑗−𝑖− 1)!

∑𝑛+𝑖 𝑘=1

(−1)^𝑘−1𝜏^𝑛−𝑘

(2𝜋𝑖)^𝑘−1(𝑛+𝑖−𝑘)!Γ^𝐿,0_{𝑛+𝑗,𝑘}(𝜏), (5.116) and thus

𝐵_𝑛,𝑟(𝜏) =𝐵_𝑛,𝑟^∞(𝜏) + (−1)^𝑛−1(2𝜋𝑖)^𝑟−1 (𝑛− 1)!

∑𝑟−1 𝑗=1

1 (𝑟−𝑗)!

∑𝑗−1 𝑖=0

(−1)^𝑖(𝑛+𝑖− 1)!

𝑖!(𝑗−𝑖− 1)! ×

×

∑𝑛+𝑖 𝑘=1

(𝑛+𝑗−𝑘− 1)!𝜏^𝑛−𝑘 (𝑛+𝑖−𝑘)!

𝑘+1∑

𝑙=2

B_{𝑘+1−𝑙}

(𝑘+ 1 −𝑙)!(2𝜋𝑖)^𝑙−1𝐴⁰_{𝑛+𝑗−𝑘,𝑙}(𝜏). (5.117) Proof. One can prove the first statement either by comparing term by term the equation of Theorem5.5.6, or by using the corollaries of Proposition5.5.1, Lemma5.5.4 and Lemma5.5.5. The second statement follows from the first plus Corollary5.5.2.

□

In particular, one can see that the range of powers of 𝜏 agrees with the range predicted by Theorem5.3.3. It is useful to write down the formulae given in the corollary, together with the explicit formula for the constant term, for the simplest case of length 2:

𝐵_𝑛,2(𝜏) = 𝐵_𝑛,2^∞(𝜏) +

∑𝑛 𝑘=1

(−1)^𝑘−1𝑛!𝜏^𝑛−𝑘

(𝑛−𝑘)!(2𝜋𝑖)^𝑘−2Γ^𝐿,0_𝑛+1,𝑘(𝜏)

= 𝐵_𝑛,2^∞(𝜏) + (−1)^𝑛−1𝑛!

∑𝑛 𝑘=1

𝜏^𝑛−𝑘

∑𝑘+1 𝑙=2

B_{𝑘+1−𝑙}

(𝑘+ 1 −𝑙)!(2𝜋𝑖)^𝑙−2𝐴⁰_{𝑛+1−𝑘,𝑙}(𝜏) The first non-trivial examples of B-elliptic MZVs (recall that in order to have interesting examples we need𝑛+𝑟odd, and we prefer to avoid B-elliptic MZVs with log(𝜏)in the expansion) given by our construction are

𝐵_3,2(𝜏) = −(2𝜋𝑖)²

720 𝜏³− 𝜁(3)

2𝜋𝑖 − 6𝜁(4) (2𝜋𝑖)²

1

𝜏 + 6𝜋𝑖𝜏²Γ^𝐿,0_4,1(𝜏) − 6𝜏Γ^𝐿,0_4,2(𝜏) + 3

𝜋𝑖Γ^𝐿,0_4,3(𝜏), 𝐵_2,3(𝜏) = (2𝜋𝑖)³

720 𝜏²−𝜋𝑖

2 𝜁(2) − 2𝜁(3)

𝜏 −3𝜁(4) 𝜋𝑖

1

𝜏² − 48𝜋²Γ^𝐿,0_4,1(𝜏) − 36𝜋𝑖Γ^𝐿,0_4,2(𝜏) +12

𝜏 Γ^𝐿,0_4,3(𝜏),

𝐵_5,2(𝜏) = (2𝜋𝑖)²

30240𝜏⁵− 𝜁(5)

(2𝜋𝑖)³ − 10𝜁(6)

(2𝜋𝑖)⁴𝜏 + 10𝜋𝑖𝜏⁴Γ^𝐿,0_6,1(𝜏)

− 20𝜏³Γ^𝐿,0_6,2(𝜏) + 30

𝜋𝑖𝜏²Γ^𝐿,0_6,3(𝜏) − 120𝜏

(2𝜋𝑖)²Γ^𝐿,0_6,4(𝜏) + 120

(2𝜋𝑖)³Γ^𝐿,0_6,5(𝜏),

𝐵_4,3(𝜏) = −(2𝜋𝑖)³ 30240𝜏⁴−1

2 𝜁(4)

2𝜋𝑖 − 4𝜁(5)

(2𝜋𝑖)²𝜏 − 20𝜁(6) (2𝜋𝑖)³𝜏²

− 40𝜋𝑖𝜏²Γ^𝐿,0_6,2(𝜏) + 120𝜏Γ^𝐿,0_6,3(𝜏) − 180

𝜋𝑖 Γ^𝐿,0_6,4(𝜏) + 480

(2𝜋𝑖)²𝜏Γ^𝐿,0_6,5(𝜏),

𝐵_3,4(𝜏) = −(2𝜋𝑖)⁴

5040 𝜏³−𝜋𝑖 𝜁(3)

6 − 3𝜁(4)

2𝜏 − 3𝜁(5)

𝜋𝑖𝜏² − 20𝜁(6)

(2𝜋𝑖)²𝜏³ +(2𝜋𝑖)³𝜏² 2 Γ^𝐿,0_4,1(𝜏)

− (2𝜋𝑖)²𝜏Γ^𝐿,0_4,2(𝜏) + 2𝜋𝑖Γ^𝐿,0_4,3(𝜏) + 120𝜋𝑖Γ^𝐿,0_6,3(𝜏) − 360

𝜏 Γ^𝐿,0_6,4(𝜏) + 360

𝜋𝑖𝜏²Γ^𝐿,0_6,5(𝜏),

𝐵_2,5(𝜏) = (2𝜋𝑖)⁵

5040 𝜏²− (2𝜋𝑖)³𝜁(2)

24 − (2𝜋𝑖)²𝜁(3)

3𝜏 −3𝜋𝑖 𝜁(4)

𝜏² −4𝜁(5)

𝜏³ − 5𝜁(6) 𝜋𝑖𝜏⁴

− (2𝜋𝑖)³Γ^𝐿,0_4,2(𝜏) + 2(2𝜋𝑖)²

𝜏 Γ^𝐿,0_4,3(𝜏) − 240𝜋𝑖

𝜏² Γ^𝐿,0_6,4(𝜏) + 480

𝜏³ Γ^𝐿,0_6,5(𝜏).

Im Dokument Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes (Seite 97-104)