Generalization of Bernoulli numbers and polynomials to the multiple case

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Generalization of Bernoulli numbers and polynomials to the multiple case

Olivier Bouillot,

Marne-la-Vall´ee University, France

Resurgence, Physics and Numbers.

May 18-22, 2015.

Centro di Ricerca Matematica Ennio De Giorgi, Pisa, May 21^st, 2015 .

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Outline

1 Introduction to mould calculus First definitions

Mould operations Primary symmetries Secondary symmetries

2 Using mould calculus: a generalization of Bernoulli numbers to the multiple case

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Outline

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Definition

Definition:

Concrete def.: A mould is a function with a variable number of variables.

Mathematical def. 1: A mould is a function defined on a monoid.

Mathematical def. 2: A mould is a collection (f0,f1,f2,· · ·) of functions,fn

being a function ofnvariables.

Typical examples: The addition : fn(x1,· · ·,xn) =x1+· · ·+xn. The Multiple Zeta Values : f0: ()7−→1

f1:s17−→ X

0<n₁

1 n^s₁¹ f2: (s1,s2)7−→ X

0<n₂<n₁

1 n^s₁¹n^s₂² ..

.

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First notations.

Let Ω be an alphabet, and Ω^?be the set of all words

New notations:

Functional notations Mould notations

Name f M^•

Evaluation f(x) M^ω^ω^ω

Why do we need some “new” notations ?

To mix easily index and exponent in notations.

To understand easily the type of object at first sight.

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Mould/comould’s contractions.

Moulds might be contracted with dual objects, calledcomoulds(which are also functions with a variable number of variables) :

Definition: Mould - Comould contraction:

The mould-comould contraction of a mouldM^•and a comouldB•is:

X

•

M^•B•:= X

ωωω∈Ω^?

M^ω^ω^ωBωωω

(if the sum is well-defined...)

For analytical reasons, a mould-comould contraction might be understood to be analgebra automorphismor aderivation.

Important remark:

All the following mould’s definitions come from such an interpretation, in particular for the mould algebra’s structure and the mould’s symmetries.

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Difference between moulds and co-moulds

Moulds: Co-moulds:

needs a commutative needs aC-algebraO, C-algebraC, in which in which co-moulds take moulds take their values their values

C= algebra of coefficients O= algebra of operators

mould = anymapΩ^?7−→C co-mould = anyhomomorphism Ω^?7−→O

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Formal mould/comould contraction

To each letterω∈Ω, we define a symbolaω, which will be, when necessary, specialized toBω.

The symbolsaω do not commute.

The symbolsaω are extended to words:

aω₁···ω_r=aω₁· · ·aω_r .

To each mouldM^•∈ M^•_C(Ω), we define a seriess(M^•)∈ChhAii, where A ={aω ; ω∈Ω}by:

s(M^•) = X

ωωω∈Ω^?

M^ω^ω^ω aωωω:=X

•

M^• a•.

Ifϕis a specialization morphism defined byϕ(aω) =Bω, then:

ϕ(s(M^•)) =X

•

M^•B•.

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Outline

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Mould operations.

LetM^•andN^• be two moulds valued in an algebraC,λ∈C.

Addition: X

•

(M^•+N^•)a•=X

•

M^•a•+X

•

N^•a•

S^•=M^•+N^• ⇐⇒ ∀ωωω∈Ω^?, S^ω^ω^ω=M^ω^ω^ω+N^ω^ω^ω. Scalar multiplication: X

•

(λM)^•a•=λX

•

M^•a•

(λM)^•=λ·M^• ⇐⇒ ∀ωωω∈Ω^?,(λM)^ω^ω^ω=λM^ω^ω^ω. Mould multiplication: X

•

(M^•×N^•)a•= X

•

M^•a•

X

•

N^•a•

P^•=M^•×N^• ⇐⇒ ∀ωωω∈Ω^?,P^ω^ω^ω= X

(ωωω1 ;ωωω2 )∈(Ω?)2 ω ω ω=ωωω1·ωωω2

M^ω^ω^ω¹N^ω^ω^ω².

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Algebraic structure

Example of mould product computation:P^•=M^•×N^• P^∅ = M^∅N^∅

P^ω¹ = M^ω¹N^∅+M^∅N^ω¹

P^ω¹^,ω² = M^ω¹^,ω²N^∅+M^ω¹N^ω²+M^∅N^ω¹^,ω² Example of operations on moulds:

A^•×B^•

×C^• = A^•× B^•×C^• . A^•×1^• = 1^•×A^•=A^•. A^•+B^•

×C^• = A^•×C^•+B^•×C^•. A^•× B^•+C^•

= A^•×B^•+A^•×C^•. A^•×B^• 6= B^•×A^•.

Property: Algebraic structure

LetM^•_C(Ω) be the set of moulds defined over the alphabet Ω and valued in the algebraC.

So, (M^•C(Ω),+, .,×) is a non commuative, associative, unitaryC-algebra.

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Outline

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Translating the Leibnitz rules

In practice, the comould satisfies some “Leibnitz rules”, which have to be translated in terms of theaω:

Bω(ϕψ) = Bω(ϕ)ψ+ϕBω(ψ) ∆(aω) = aω⊗1 + 1⊗aω

Bω(ϕψ) = X

ω1,ω2∈Ω ω1 +ω2 =ω

Bω1(ϕ)Bω2(ψ) ∆(aω) = X

ω1,ω2∈Ω ω1 +ω2 =ω

aω1⊗aω2

(where (Ω,+) is a semi-group)

∆ is extended to words over A in order to be an algebra homomorphism, and then toChhAiiby linearity. So,

(ChhAii,·,∆) is a Hopf algebra.

Property:

Letϕ:aω7−→Bω be a specialization morphism.

1 If ∆ (s(M^•)) =s(M^•)⊗s(M^•) (i.e. is group-like), thenϕ(s(M^•)) is an automorphism.

2 If ∆ (s(M^•)) =s(M^•)⊗1 + 1⊗s(M^•) (i.e. is primitive), thenϕ(s(M^•)) is a derivation.

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The symmetrAl case: ∆(a

ω

) = a

ω

⊗ 1 + 1 ⊗ a

ω

1/3

Definition: shuffle product

Foru,v∈Ω^? anda,b∈Ω, we define recursively the shuffle productby:

1u = u1 = u.

uavb = (uvb)a+ (uav)b

LetshA(ωωω¹;ωωω²) be the set of words which appear inωωω¹ωωω², counted with their multiplicity.

Lemma:

For any wordωωω∈Ω^?, ∆(ωωω) = X

ω ω ω1,ωωω2∈Ω? ωω

ω∈shA(ωωω1,ωωω2 )

ωωω1⊗ωωω2.

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The symmetrAl case: ∆(a

ω

) = a

ω

⊗ 1 + 1 ⊗ a

ω

2/3

Property:

A mouldM^•∈ M^•_C(Ω) satisfies:

1 ∆(s(M^•)) =s(M^•)⊗s(M^•) if, and only if, the mouldM^• satisfies:

∀(ωωω¹;ωωω²)∈(Ω^?)², X

ω ω

ω∈shA(ωωω¹;ωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω=M^ω^ω^ω¹M^ω^ω^ω² . (1)

2 ∆(s(M^•)) =s(M^•)⊗1 + 1⊗s(M^•) if, and only if, the mouldM^•satisfies:







M^∅= 0.

∀(ωωω¹;ωωω²)∈(Ω^?− {∅})², X

ωω

ω∈shA(ωωω¹;ωωω²)

M^ω^ω^ω= 0 . (2)

Sketch of proof of(1): ∆(s(M^•)) = X

ω∈Ω^?

M^ω∆(ω) = X

(ω,ω1,ω2 )∈(Ω?)3 ω∈shA(ω1,ω2 )

M^ωω1⊗ω2

⇓

∆(s(M^•))−s(M^•)⊗s(M^•) = X

(ω₁,ω₂)∈(Ω^?)²

X

ω∈Ω ω∈shA(ω1,ω2 )

M^ω−M^ω¹M^ω²

ω1⊗ω2.

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The symmetrAl case: ∆(a

ω

) = a

ω

⊗ 1 + 1 ⊗ a

ω

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Definition

A mouldM^•∈ M^•C(Ω) is called:

1 symmetrAl when:

∀(ωωωωωωωωω¹;ωωωωωωωωω²)∈(Ω^?)², X

ωωωωωωωωω∈shA(ωωωωωωωωω¹;ωωωωωωωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω=M^ω^ω^ω^ω^ω^ω^ω^ω^ω¹M^ω^ω^ω^ω^ω^ω^ω^ω^ω² . (3)

2 alternAl when:







M^∅= 0.

∀(ωωωωωωωωω¹;ωωωωωωωωω²)∈(Ω^?− {∅})², X

ω ωω

ωωωωωω∈shA(ωωωωωωωωω¹;ωωωωωωωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω= 0 . (4)

Reformulation of the previous property:

A mouldM^•∈ M^•_C(Ω) is:

1 symmetrAl if, and only if, ∆(s(M^•)) =s(M^•)⊗s(M^•).

2 alternAl if, and only if, ∆(s(M^•)) =s(M^•)⊗1 + 1⊗s(M^•).

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An example of symmetrAl and alternAl moulds

Example 1:

Let Ω be a set, and for allω∈Ω, letfΩ: [0; 1]7−→Rbe a continuous function.

LetI^•be the mould defined by:

I^ω¹^,···^,ω^r = Z

0<tr<···<t₁<1

fω₁(t1)· · ·fω_r(tr)dt1· · ·dtr.

Then,I^•is symmetrAl .

Example 2:

Any mouldM^•∈ M^•_C(Ω) such that:

l(ω)6= 1 =⇒M^ω= 0, is alternAl .

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The symmetrEl case: ∆(a

ω

) = X

ω1,ω2∈Ω ω1⊥ω2=ω

a

ω1

⊗ a

ω2

1/3

Definition: stuffle product

Suppose that (Ω,+) is an additive semi-group. Foru,v ∈Ω^?anda,b∈Ω, we define recursively the stuffle product by:

ε u = u ε = u .

ua vb = (u vb)a+ (ua v)b+ (u v)(a+b). (5)

LetshE(ωωω¹;ωωω²) be the set of words which appear inωωω¹ ωωω², counted with their multiplicity.

Lemma:

For any wordωωω∈Ω^?, ∆(ωωω) = X

ω ωω1,ωωω2∈Ω? ωω

ω∈shE(ωωω1,ωωω2 )

ωωω1⊗ωωω2.

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The symmetrEl case: ∆(a

ω

) = X

a

ω1

⊗ a

ω2

2/3

Property:

A mouldM^•∈ M^•_C(Ω) satisfies:

1 ∆(s(M^•)) =s(M^•)⊗s(M^•) if, and only if, the mouldM^• satisfies:

∀(ωωω¹;ωωω²)∈(Ω^?)², X

ωωω∈shE(ωωω¹;ωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω=M^ω^ω^ω¹M^ω^ω^ω² . (6)

2 ∆(s(M^•)) =s(M^•)⊗1 + 1⊗s(M^•) if, and only if, the mouldM^•satisfies:







M^∅= 0.

∀(ωωω¹;ωωω²)∈(Ω^?− {∅})², X

ωω

ω∈shE(ωωω¹;ωωω²)

M^ω^ω^ω= 0. (7)

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The symmetrEl case: ∆(a

ω

) = X

a

ω1

⊗ a

ω2

3/3

Definition

A mouldM^•∈ M^•_C(Ω) is called:

1 symmetrEl when:

∀(ωωωωωωωωω¹;ωωωωωωωωω²)∈(Ω^?)², X

ω ω ω ω ωω

ωωω∈shE(ωωωωωωωωω¹;ωωωωωωωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω=M^ω^ω^ω^ω^ω^ω^ω^ω^ω¹M^ω^ω^ω^ω^ω^ω^ω^ω^ω² . (8)

2 alternEl when:







M^∅= 0.

∀(ωωωωωωωωω¹;ωωωωωωωωω²)∈(Ω^?− {∅})², X

ωωωωωωωωω∈shE(ωωωωωωωωω¹;ωωωωωωωωω²)

M^ω^ω^ω^ω^ω^ω^ω^ω^ω= 0. (9)

Reformulation of the previous property:

A mouldM^•∈ M^•_C(Ω) is:

1 symmetrEl if, and only if, ∆(s(M^•)) =s(M^•)⊗s(M^•).

2 alternEl if, and only if, ∆(s(M^•)) =s(M^•)⊗1 + 1⊗s(M^•).

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An example of symmetrEl and alternEl moulds

Example 1:

Let (un)n∈N≥1 be a sequence of complex numbers such that:

∃α≥2, un =

n−→+∞O(n^−α). LetS^•∈ M^•_R(N^≥1) be the mould defined by:

S^s¹^,···^,s^r = X

0<n_r<···<n₁<+∞

un^s¹₁· · ·un^s²_r.

Then,S^•is symmetrEl . Example 2:

The mouldM^•∈ M^•_R(Ω) defined by:

M^ω=(−1)^l(ω) l(ω) , is alternEl .

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Outline

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Formal moulds

Other point of view on moulds: A mould is a collection of functions (f0,f1,f2,· · ·), wherefi : Ωⁱ 7−→C.

Definition:

A formal mould is a collection of formal series (S0,S1,S2,· · ·), whereSnis a formal power series innindeterminates (and consequently,S0is constant)

Notation:F M^•_C={formal mould with values in the algebraC}.

What is the difference between a mould and a formal mould?

MouldM^•∈ M^•C(Ω) Formal mouldM^•∈ F M^•C

where Ω = (X1,X2,· · ·).

No link betweenM^X¹^,X² andM^X²^,X¹ !!! M^X¹^,X² andM^X²^,X¹ are related by thesubstitutionof the indeterminates.

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Generics example of formal moulds

Definition:

With a mouldM^•∈ M^•C(N), we associate two formal mouldsMog^•andMeg^• defined by:











Mog^X¹^,···^,X^r = X

s₁,···,sr∈N

M^s¹^,···^,s^rX₁^s¹· · ·Xr^s^r .

Meg^X¹^,···^,X^r = X

s₁,···,sr∈N

M^s¹^,···^,s^rX₁^s¹ s1! · · ·X_r^s^r

sr! .

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Secondary symmetries

Definition:

The secondary symmetries of a mould are the symmetries of its corresponding formal moulds.

Theorem: (Ecalle,∼80⁰s) LetM^•∈ M^•_C(N) be a mould.

1 M^•is symmetrAl if, and only if,Mog^•is symmetrAl .

2 M^•is alternAl if, and only if,Mog^• is alternAl .

Theorem: (B., 2015)

LetM^•∈ M^•_C(N) be a mould.

1 M^•is symmetrAl if, and only if,Meg^• is symmetrAl .

2 M^•is alternAl if, and only if,Meg^•is alternAl .

3 M^•is symmetrEl if, and only if,Meg^•is symmetrEl .

4 M^•is alternEl if, and only if,Meg^• is alternEl .

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Algebraic construction to prove our result on secondary symmetries

1 Let us consider:

X={X1,X2,· · · }be an infinite set of indeterminates.

Y=NX= (

X

x∈X

λxx ; (λX)X∈X∈N^Xhas finitely nonzero terms )

. A ={Ay ; y ∈Y}.

2 We extend the definition ofMog^•andMeg^•to words constructed over A^?: Mog^A^y¹^,···^,A^yr :=Mog^y¹^,···^,y^r , Meg^A^y¹^,···^,A^yr :=Meg^y¹^,···^,y^r

3 We define a secondary formal mould/comould contraction,i.e. to a formal mouldFM^•∈ F M^•_C, we associate a seriesS(FM^•)∈C[[X]]hhAii by:

S(FM^•) = X

ωωω∈A^?

FM^ω^ω^ωAωωω:=X

•

FM^•A•.

4 We define a map ∆ as before,i.e. a coproduct, by:

Case Al: ∆(Ay) =Ay⊗1 + 1⊗Ay ∆(AAA) = X

BBB,CCC∈A? A A A∈shA(BBB,CCC)

BBB⊗CCC

Case El: ∆(Ay) = X

u,v∈Y u+v=y

Au⊗Av ∆(AAA) = X

BB B,CCC∈A? A A A∈shE(BBB,CCC)

B B B⊗CCC

5 Now, we just have to adapt the proof of

∆(s(M^•)) =s(M^•)⊗s(M^•)⇐⇒M^•is symmetrAl /symmetrEl .

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Algebraic construction to prove our result on secondary symmetries

1 Let us consider:

Y=NX= (

X

x∈X

. A ={Ay ; y ∈Y}.

S(FM^•) = X

ωωω∈A^?

•

FM^•A•.

BBB⊗CCC

u,v∈Y u+v=y

B B B⊗CCC

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Algebraic construction to prove our result on secondary symmetries

1 Let us consider:

Y=NX= (

X

x∈X

. A ={Ay ; y ∈Y}.

S(FM^•) = X

ωωω∈A^?

•

FM^•A•.

BBB⊗CCC

u,v∈Y u+v=y

B B B⊗CCC

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Algebraic construction to prove our result on secondary symmetries

1 Let us consider:

Y=NX= (

X

x∈X

. A ={Ay ; y ∈Y}.

S(FM^•) = X

ωωω∈A^?

•

FM^•A•.

BBB,CCC∈A? AA A∈shA(BBB,CCC)

BBB⊗CCC

u,v∈Y u+v=y

B B B⊗CCC

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Algebraic construction to prove our result on secondary symmetries

1 Let us consider:

Y=NX= (

X

x∈X

. A ={Ay ; y ∈Y}.

S(FM^•) = X

ωωω∈A^?

•

FM^•A•.

BBB,CCC∈A? AA A∈shA(BBB,CCC)

BBB⊗CCC

u,v∈Y u+v=y

B B B⊗CCC

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Exercices

1 Adapt the previous construction to have a better understanding of the symmetrIlity . Then prove that:

a. M^•is symmetrEl⇐⇒Mog^•is symmetrIl . b. M^•is alternEl⇐⇒Mog^•is alternIl .

2 Letδa,bbe the Kronecker symbol.

LetM^•be a mould such that:











M^aM^b = M^a,b+M^b,a+δa,bM^b

MâM^b,c = Mâ,b,c+M^b,a,c+M^b,c,a+δa,bMâ,c+δa,cM^b,a

Mâ,bM^c,d = Mâ,b,c,d+Mâ,c,b,d+Mâ,c,d,b+M^c,d,a,b+M^c,a,d,b+M^c,a,b,d +δa,c(M^c,b,d+M^c,d,b) +δa,cδb,dM^c,d

.. .

a. Define a (primary) mould symmetry satisfied byM^•. b. What are its corresponding secondary symmetries ?

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Outline

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Why multiple Bernoulli numbers and polynomials?

Definition:

The numbersZe^s¹^,···^,s^r defined by Ze^s¹^,···^,s^r = X

0<n_r<···<n₁

1 n1s₁· · ·nrs_r ,

wheres1,· · ·,sr ∈Csuch that<(s1+· · ·+sk)>k,k∈[[1;r]], are called multiple zeta values.

Fact:There exist at least three different ways to renormalise multiple zeta values at negative integers.

Ze_MP^0,−2(0) = 7

720 , Ze_GZ^0,−2(0) = 1

120 , Ze^0,−2_FKMT(0) = 1 18 . Question:Is there a group acting on the set of all possible multiple zeta values renormalisations?

Main goal:Define multiple Bernoulli numbers in relation with this.

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On the Riemann and Hurwitz Zeta Functions

Definition:

The Riemann and Hurwitz zeta functions are defined, for<e s>1, and z∈C−N^<0, by:

ζ(s) =X

n≥0

1

n^s , ζ(s,z) =X

n≥0

1 (n+z)^s .

Property:

s7−→ζ(s) ands7−→ζ(s,z) can be analytically extend to a meromorphic function onC, with a simple pole located at 1.

Remark: ζ(−n) = −bn+1

n+ 1 for alln∈N. ζ(−n,z) = −Bn+1(z)

n+ 1 for alln∈N.

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Hurwitz Multiple Zeta Functions

Definition: Hurwitz multiple zeta functions

Letz∈C−N<0and (s1,· · ·,sr)∈(N≥1)^r, such thats1≥2 . The Hurwitz multiple zeta functions are defined by:

He^s¹^,···^,s^r(z) = X

0<nr<···<n₁

1

(n1+z)^s¹· · ·(nr+z)^s^r .

Lemma: (B., Ecalle, 2013)

Let ∆−be the difference operator: ∆−(f)(z) =f(z−1)−f(z) Then: ∆−(He^•) =He^•×J^•,whereJ^s¹^,···s^r(z) =

z^−s¹ ifr = 1.

0 otherwise.

Heuristic:

Be^s¹^,···^,s^r(z) = Multiple (divided) Bernoulli polynomials =He^−s¹^,···^,−s^r(z). be^s¹^,···^,s^r= Multiple (divided) Bernoulli numbers =He^−s¹^,···^,−s^r(0).

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Some Properties of Bernoulli Polynomials

Property 1: Difference Equation

∆ Bn

n

(x) =xⁿ⁻¹for alln∈N^∗, where ∆(f)(z) =f(z+ 1)−f(z).

Property 2: Reflexion Formula

(−1)ⁿBn(1−x) =Bn(x) for alln∈N.

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Main Goal

We want to define a mouldBe^•(z)∈ M^•_Q_[z](N) such that:

its values have properties similar to Hurwitz Multiple Zeta Functions’

properties.

its values have properties generalizing these of Bernoulli polynomials.

Main Goal:

FindBe^•∈ M^•_C_[z](N) such that:











Be^s(z) =Bs+1(z)

s+ 1 , wheres≥0 ;

∆ (Be^•) (z) =Be^•×J^•(z) Be^•is symmetrEl .

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A singular solution











Be^s(z) = Bs+1(z)

s+ 1 , wheres≥0

∆ (Be^•) (z) =Be^•×J^•(z) Be^•is symmetrEl .

⇐⇒











Beeg^X(z) = e^zX e^X−1− 1

X

∆ (Beeg^•) (z) =Beeg^•×Jeg^•(z) Beeg^•is symmetrEl .

Reminders:

1 Jeg^X(z) =e^zX andJeg^X¹^,···^,X^r = 0 ifr6= 1.

2 ∆

e^zX e^X−1

=e^zX .

From a false solution to a singular solution...

Sing^X¹^,···^,X^r(z) = _r ^e^z(^Xk¹^+···+^Xkr⁾ Y

i=1

(e^X^k¹^+···+X^ki −1)

is a false solution to the system we are

solving...:

Sing^•(z)6∈ F M^•_C[X]butSing^•(z)∈ F M^•_C[[X]].

Heuristic:Find a correction ofS, to send it intoC[z][[X]]hhAii.

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Another solution

Fact:If ∆(f)(z) =f(z−1)−f(z), ker ∆∩zC[z] ={0}.

Consequence:There exists a unique family of polynomials (Be^s¹^,···^,s^r) such that:

( Be₀ⁿ¹^,···^,n^r(z+ 1)−Be₀ⁿ¹^,···^,n^r(z) =Be₀ⁿ¹^,···^,n^r−1(z)zⁿ^r . Be₀ⁿ¹^,···^,n^r(0) = 0.

This producesBe0^•∈ M^•_C_[z](N) andBeeg0^•∈ F M^•_C_[z]. Lemma: (B., 2013)

1 The mouldsBe0^•(z) andBeeg0^•(z) are symmetrEl .

2 The mouldBeeg0^•(z) satisfies a recurence relation:











Be₀^Y¹(z) = e^zY¹−1 e^Y¹−1

Be₀^Y¹^,···^,Y^r(z) =Be₀^Y¹^+Y²^,Y³^,···^,Y^r(z)− Be₀^Y²^,Y³^,···^,Y^r(z) e^Y¹−1

3 Beeg0^•(z) = Sing^•(0)^×−1

× Sing^•(z).

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Characterization of the set of solutions

Proposition: (B. 2013)

Any familly of polynomials which are solution of the previous system comes from a formal mouldBeeg^•valued inC[z] such that there exists a formal mouldbeeg^•valued inCsatisfying:

1.beeg^X = 1 e^X −1− 1

X 2. beeg^•is symmetrEl 3.Beeg^•(z) =beeg^•× Beeg0^•(z) =beeg^•× Sing^•(0)×−1

× Sing^•(z) .

Theorem: (B., 2013)

The subgroup of symmetrEl formal moulds with vanishing coefficients in length 1, acts on the set of all possible multiple Bernoulli polynomials,i.e. on the set of all possiblealgebraicrenormalization of multiple zeta values at negative integers.

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Using the Reflexion Formula of Bernoulli polynomials

Notations: M^−•: (ω1,· · ·, ωr)7−→M^−ω¹^,···^,−ω^r . M

←−•

: (ω1,· · ·, ωr)7−→M^ω^r^,···^,ω¹ .

Lemma: (B., 2013)

LetSg^•be the mould defined by: Sg^X¹^,···^,X^r = (−1)^r. For allz∈C, we have:

Sing^−•(0) = Sing

←−•

(0)^×−1

× Sg^• , Sing^−•(1−z) = Sing

←−•

(z)^×−1 .

Sketch of proof:Use recursively 1

e^X−1+ 1

e^−X −1+ 1 = 0 . Examples:

Sing^−X,−Y(0) = Sing^X^,Y(0) +Sing^X+Y(0) +Sing^X(0) + 1 . Sing^−X,−Y(1−z) = Sing^X^,Y(z) +Sing^X^+Y(z).

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Reflexion Formula for Multiple Bernoulli Polynomials

Theorem: (B., 2013) For allz∈C, we have:

Beeg^−•(1−z)× Beeg

←−•

(z) =beeg^−•×(1^•+I^•)×beeg

←−•

, where 1^X¹^,···^,X^r =

1 ifr= 0.

0 ifr>0. and I^X¹^,···^,X^r =

1 ifr= 1. 0 ifr6= 1. Heuristic:

Sing^−•(1−z)× Sing

←−•

(z) = 1^•

Consequently, we will definebeeg^•such that:

beeg^−•×(1^•+I^•)×beeg

←−•

= 1^•

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Resolution of a mould equation

Consequently, we will definebeeg^•such that:

beeg^−•×(1^•+I^•)×beeg

←−•

= 1^•

⇐⇒ beeg^−•×(Sg^•)^×−1×beeg

←−•

= 1^•

⇐⇒ ceeg^−•×ceeg

←−•

= 1^•,whereceeg^•=beeg^•×(Sg^•)^×−¹²

⇐⇒ Exp(deg^−•)×Exp(deg

←−•

) = 1^•,whereceeg^•=Exp(deg^•)

⇐⇒ deg^−•+deg

←−•

=O^•.

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Definition of Multiple Bernoulli Polynomials and Numbers

The moulddeg^•defined by







deg^∅:= 0,

deg^X¹^,···^,X^r :=(−1)^r−1

r deg^X¹^+···+X^r , wheredeg^X = 1

e^X−1− 1 X +1

2 , satifies the required conditions:

(

deg^•+deg

←−−•

= 0, deg^•is alternEl.

Definition: (B., 2014)

The mouldsBeeg^•(z) andbeeg^•are defined by:

Beeg^•(z) = Exp(deg^•)×√

Sg^•×(Sing^•(0))^×−1× Sing^•(z). beeg^• = Exp(deg^•)×√

Sg^•

The coefficientsBe^•(z) andBe^•of these exponential generating series are called respectively theMultiple Bernoulli Polynomials and Numbers.

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Examples of explicit expression for multiple Bernoulli numbers

Consequently, we obtain explicit expressions like, forn1,n2,n3>0:

beⁿ¹^,n² = 1 2

bn₁+1

n1+ 1 bn₂+1

n2+ 1− bn₁+n₂+1

n1+n2+ 1

.

beⁿ¹^,n²^,n³ = +1 6

bn₁+1

n1+ 1 bn₂+1

n2+ 1 bn₃+1

n3+ 1

−1 4

bn₁+n₂+1

n1+n2+ 1 bn₃+1

n3+ 1+ bn₁+1

n1+ 1

bn₂+n₃+1

n2+n3+ 1

+1 3

bn₁+n₂+n₃+1

n1+n2+n3+ 1 .

Remark:Ifn1= 0,n2= 0 orn3= 0, the expressions are not so simple...

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Table of Multiple Bernoulli Numbers in length 2

be^p,q p= 0 p= 1 p= 2 p= 3 p= 4 p= 5 p= 6 q= 0 3

8 −1

12 0 1

120 0 − 1

252 0

q= 1 −1 24

1 288

1

240 − 1

2880 − 1 504

1 6048

1 480

q= 2 0 1

240 0 − 1

504 0 1

480 0

q= 3 1

240 − 1

2880 − 1 504

1 28800

1

480 − 1

60480 − 1 264

q= 4 0 − 1

504 0 1

480 0 − 1

264 0

q= 5 − 1 504

1 6048

1

480 − 1

60480 − 1 264

1 127008

691 65520 one out of four Multiple Bernoulli Numbers is null.

one out of two antidiagonals is “constant”.

“symmetry” relatively top=q.

cross product around the zeros are equals : 28800·127008 = 60480² .

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Conclusion

1. We have respectively defined the Multiple Bernoulli Polynomials and Multiple Bernoulli Numbers by:







Beeg^•(z) = Exp(deg^•)×√ Sg^•×

Sing^•(0)×−1

× Sing^•(z). beeg^• = Exp(deg^•)×√

Sg^•

2. The Multiple Bernoulli Polynomials satisfy a nice generalization of:

the nullity ofb2n+1ifn>0:

b^2p¹^,···^,2p^r = 0. the symmetryBn(1) =Bn(0) ifn>1:

ifnr >0, Bⁿ¹^,···^,n^r(0) =Bⁿ¹^,···^,n^r(1). the difference equation ∆(Bn)(x) =nxⁿ⁻¹:

Bⁿ¹^,···^,n^r(z+ 1)−Bⁿ¹^,···^,n^r(z) =Bⁿ¹^,···^,n^r−1(z)·zⁿ^r . the reflection formula (−1)ⁿBn(1−x) =Bn(x):

Beeg^−•(1−z)· Beeg

←−•

(z) = 1^•.

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Generalization of Bernoulli numbers and polynomials to the multiple case