Mould calculus, resurgence and combinatorial Hopf algebras.

Mould calculus, resurgence and combinatorial Hopf algebras.

Olivier Bouillot The 15 november 2011.

Table des mati` eres

1 Aim. 3

2 First expression of the horn map. 4

3 Evaluation of Γ _• · z. 5

4 Multitangent functions / Multizetas values. 6

5 Permutation of the two sums. 8

6 Calculation of Fourier coefficients of h ⁺ − id. 9

During this talk, we set :

• f (z) = z + 1 + a(z) where a(z) = X

n≥3

a _n

z ⁿ⁻¹ ∈ 1 z ² C

1 z

:

∃(C ₀ ; C ₁ ) ∈ ( R ⁺ ∗ ) ² , |a _n | ≤ C ₀ C ₁ ⁿ .

• t(z) = z + 1 .

• g(z) = f (z) − 1, so f = l ◦ g .

• N p = {n ∈ N ; n ≥ p}, where p ∈ N .

• seq(Ω) = {ω = (ω ₁ ; · · · ; ω _r ) ; r ∈ N ^∗ , (ω ₁ , · · · , ω _r ) ∈ Ω ^r } , where Ω is an alphabet.

• ∀s = (s ₁ , · · · , s _r ) ∈ seq( N ),

||s|| = s ₁ + · · · + s _r

l(s) = r

1 Aim.

We want to proof the following result :

Theorem : Let A _• defined by : ∀s ∈ seq( N 3 ) , A _s = a _{s 1} · · · a _{s r} . A _• defined by : ∀S ∈ seq seq( N ³ ) − {∅}

, A _S = A _s ¹ · · · A _s ^r .

then :

1. there exists an explicit mould τ ^• defined on seq seq( N ³ ) − {∅})

such that :

π ⁺ = X

S∈seq(seq( N 3 )−{∅})

sg ^S τ ^S A _S .

2. for all n ∈ Z ^∗ , there exists an explicit mould τ b _n ^• , defined on seq seq( N 3 ) −{∅}

such that :

∀n ∈ Z ^∗ , A ⁺ _2inπ = sg(n) X

S∈seq(seq( N 3 )−{∅})−{∅}

sg ^S τ b _n ^S A _S .

Remark : For all S ∈ seq seq( N 3 ) − {∅}

:







τ ^S ∈ MTGF ⊂ H( C − Z ) .

τ b _n ^S ∈ MZV .

2 First expression of the horn map.

We have : h ⁺ = v ^att ◦ (v ^rep ) ⁻¹ . In David’s talk, we have seen that :

v ^att (z) = z + X

k∈seq( N 1 )

ψ ^k (z) .

We can do the same with V ^att (ϕ) = ϕ ◦ v ^att , the substitution operator associate to v ^att . We obtain :

V ^att = X

n∈seq( Z )

U ⁿ ₊ Γ _n ,

where :



 

 

 

 

Γ _n = T ⁿ ◦ (G − Id) ◦ T ⁻ⁿ Γ _n = Γ _{n r} ◦ · · · ◦ Γ _{n 1}

U ⁿ ₊ =

1 if 0 ≤ n _r < · · · < n ₁ 0 else

By the same (V ^rep ) ⁻¹ = X

n∈seq( Z )

U ⁿ ₋ Γ _n where : U ⁿ ₋ =

1 if n _r < · · · < n ₁ < 0 0 else

So : H ⁺ =





X

n∈seq( Z )

U ⁿ ₊ Γ _n



 ◦





X

n∈seq( Z )

U ⁿ ₋ Γ _n





= X

U ^• ₊ × U ^• ₋ n

Γ _n

3 Evaluation of Γ _• · z.

Cf. transparents

4 Multitangent functions / Multizetas values.

Let us make a little break with special functions...

Definition : Let S ₊ ^? =

s ∈ seq( N ^∗ ); s ₁ ≥ 2 S ^? =

s ∈ seq( N ^∗ ); s ₁ ≥ 2 et s _l(s) ≥ 2

So ∀s ∈ S ₊ ^? , Ze ^s = X

1≤n r <···<n ₁

1 n ^s ₁ ¹ · · · n ^s r ^r

.

∀s ∈ S ^? , T e ^s = X

−∞<n _r <···<n ₁ <+∞

1

(n ₁ + z) ^{s 1} · · · (n _r + z) ^{s r} . Why call we “multitangents functions” ?

Because T e ¹ = π

tan(πz) ...

So the “multicotangents functions” · · · Notation : MZV = Vect _Q (Z e ^s ) _{s∈S +} ^? .

MTGF = Vect _Q (T e ^s ) _s∈S ^? .

Remark : 1. Ze ^• and T e ^• look simillar.

Proposition 1 : ∀s ∈ S ^? , ∃(z _k ) _2≤k≤m ∈ MZV ^m−1 , T e ^• =

m

X

k=2

z _k T e ^k

Proof : 1. Partial fraction decomposition of 1

(n ₁ + X) ^s

· · · (n _r + X) ^s

. give :

T e ^• =

m

X

k=1

z k T e ^k

2. z 1 = 0 because of iterate integral representation of multizetas values.

Proposition 2 : For all s ∈ S ^? , we are able to calculate Fourier coefficients of T e ^s .

Proof : From Proposition 1, it suffice to evaluate Fourier coefficient of monotangent function.

T e ^s = (−1) ^s−1 (s − 1)!

∂ ^s−1

∂z ^s−1

π tan(πz)

We know how to do for z 7−→ π tan(πz) It’s OK

Everything here is explicit !

5 Permutation of the two sums.

Come back to the horn map h ⁺ : We have :

h ⁺ (z) − z = X

r≥1

X

n∈seq( Z )

X

S ∈ seq(seq( N 3 )−{∅}) l(S)=l(n)

l(s ¹ )=1

sg ^S





X

i∈triangle(S)

B ^S,i





r

Y

p=1

1

(z + n _p ) ^{||s p ||−l(s p )+diag p i}







A _S

So, we would like :

π ⁺ (z) − z = X

r≥1

X

S ∈ seq(seq( N 3 )−{∅}) l(S)=r

l(s ¹ )=1

sg ^S





X

i∈triangle(S)

B ^S,i T e ^σ(S;i)



A _S ,

where σ(S; i) = (||s ^p || − l(s ^p ) + diag _p ⁱ ) _1≤k≤r

= X

r≥1

X

S ∈ seq(seq( N 3 )−{∅}) l(S)=r

sg ^S τ ^S A _S ,

Here :

 id , if S = ∅

6 Calculation of Fourier coefficients of h ⁺ − id.

Since h ⁺ (z) − z = X

seq( Z )

U ^• Γ _• · z is normally convergent on every

compact