Multiple Bernoulli Polynomials and Numbers

Home , Bernoulli polynomials

Olivier Bouillot, Marnes-la-Vall´eeUniversity, France

72th Sminaire Lotharingien de Combinatoire, Lyon, Wenesday 26th March 2014 . Outline

1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function

2 The Form of a Multiple Bernoulli Polynomial

3 The General Reﬂexion Formula of Multiple Bernoulli Polynomial

4 An Example of Multiple Bernoulli Polynomial Two Equivalent Deﬁnitions of Bernoulli Polynomials / Numbers

Bernoulli numbers: Bernoulli polynomials :

By a generating function: By a generating function:

t X tn text X tn = b . = B (x) . et − 1 n n! et − 1 n n! n≥0 n≥0

By a recursive formula: By a recursive formula:

  b = 1 , B0(x) = 1 ,  0  0  n  ∀n ∈ N , Bn+1(x) = (n + 1)Bn(x) , X n + 1 Z 1 ∀n ∈ N , bk = 0 .  k  Bn(x) dx = 0 .  k=0  0

First examples: First examples:

1 1 1 1 bn = 1, − , , 0, − , 0, , ··· B0(x) = 1 , 2 6 30 42 1 B (x) = x − , 1 2 1 B (x) = x 2 − x + , 2 6 . . Some Properties of Bernoulli Polynomials

Property 1: Diﬀerence Equation n−1 ∗ ∆(Bn)(x) = nx for all n ∈ N , where ∆(f )(z) = f (z + 1) − f (z).

Property 2: Reﬂexion Formula n (−1) Bn(1 − x) = Bn(x) for all n ∈ N .

Property 3: Expression in the falling factorial basis n X n + 1 n B (x) − b = (x) for all n ∈ , where: n+1 n+1 k + 1 k k+1 N k=0 • (x)n = x(x − 1) ··· (x − n + 1) . n • denotes the Stirling number of second kind. k On the Riemann Zeta Function and Hurwitz Zeta Function

Deﬁnition: The Riemann Zeta Function and Hurwitz Zeta Function are deﬁned, for 1, and z ∈ C − N<0, by: X 1 X 1 ζ(s) = , ζ(s, z) = . ns (n + z)s n≥0 n≥0

Property: s 7−→ ζ(s) and s 7−→ ζ(s, z) can be analytically extend to a meromorphic function on C, with a simple pole located at 1.

b Remark: ζ(−n) = − n+1 for all n ∈ . n + 1 N B (z) ζ(−n, z) = − n+1 for all n ∈ . n + 1 N Multiple Zeta Values and Hurwitz Multiple Zeta Functions

Deﬁnition of Hurwitz Multiple Zeta Functions

s1,··· ,sr X 1 He (z) = s s , if z ∈ C − N<0 and (n1 + z) 1 ··· (nr + z) r 0

Lemma:

For all sequences (s1, ··· , sr ), we have:

s1,··· ,sr s1,··· ,sr−1 sr ∆−(He ) = He · J ,

s −s where J (z) = z and ∆−(f )(z) = f (z − 1) − f (z).

Heuristic:

Bes1,··· ,sr (z) = Multiple (Divided) Bernoulli Polynomials = He−s1,··· ,−sr (z) .

bes1,··· ,sr = Multiple (Divided) Bernoulli Numbers = He−s1,··· ,−sr (0) . Outline

1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function

2 The Form of a Multiple Bernoulli Polynomial

3 The General Reﬂexion Formula of Multiple Bernoulli Polynomial

4 An Example of Multiple Bernoulli Polynomial Main Goal

We want to deﬁne Bes1,··· ,sr (z) such that: their properties are similar to Hurwitz Multiple Zeta Functions’ properties. their properties generalize these of Bernoulli polynomials.

Main Goal: Find some polynomials Bes1,··· ,sr such that:

 s Bs+1(z)  Be (z) = , where s ≥ 0 ;  s + 1   s1,··· ,sr s1,··· ,sr−1 sr ∆ (Be )(z) = Be (z) · z , where r ≥ 2 , and s1, ··· , sr ≥ 0 ;     the Bes1,··· ,sr are multiplied by the stuﬄe product. A particular solution of the main goal

Lemma: The familly of diﬀerence equations

∆ (Bes1,··· ,sr )(z) = Bes1,··· ,sr−1 (z) · z sr

s1,··· ,sr has a unique familly of solutions Be0 ∈ zC[z] , which are multiplied by the stuﬄe.

Sketch of proof: ker ∆ ∩ zC[z] = {0} .

Examples: p zX X p X e − 1 Beeg X (z) = Be (z) = . 0 0 p! eX − 1 p≥0 p q X +Y Y X p,q X Y Beeg (z) − Beeg (z) Beeg X ,Y (z) = Be (z) = 0 0 . 0 0 p! q! eX − 1 p,q≥0 The Form of a Multiple Bernoulli Polynomial

s1,··· ,sr • Notations: The previous family Be0 is denoted by Be0 . In general, a family Ms1,··· ,sr is denoted by M• and called a mould .

r s ,··· ,s X s ,··· ,s s ,··· ,s Mould product: M• = A• × B• ⇐⇒ M 1 r = A 1 k B k+1 r . k=0

Theorem: • The solutions Be (z), valued in C[z] and multiplied by the stuﬄe product, to the diﬀerence equation ∆ (Bes1,··· ,sr )(z) = Bes1,··· ,sr−1 (z) · z sr satisfying B (z) Bes (z) = s+1 have an exponential generating function Beeg •(z) of the s + 1 form

• • • Beeg (z) = beeg × Beeg0 (z) , where:  • beeg is valued in C .   The beeg •’s are multiplied by the stuﬄe product. 1 1  beeg X = − .  eX − 1 X Outline

1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function

2 The Form of a Multiple Bernoulli Polynomial

3 The General Reﬂexion Formula of Multiple Bernoulli Polynomial

4 An Example of Multiple Bernoulli Polynomial • A New Expression of the Mould Beeg0 (z)

New Goal: • • • • From Beeg (z) = beeg × Beeg0 (z), determine a suitable mould be , or beeg • such that the reﬂexion formula is a nice generalization of

n (−1) Bn(1 − z) = Bn(z) , n ∈ N .

Deﬁnition: z(X1+···+Xr ) X ,··· ,X e Sing 1 r (z) = , for all z ∈ C. (eX1 − 1)(eX1+X2 − 1) ··· (eX1+···+Xr − 1)

Property: ×−1 • • • Beeg0 (z) = Sing (0) × Sing (z) . Key Point of the Reﬂexion Formula of Sing •(z)

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

←− • ωr ,··· ,ω1 M :(ω1, ··· , ωr ) 7−→ M .

Lemma: Let Sg • be the mould deﬁned by: Sg X1,··· ,Xr = (−1)r . For all z ∈ C, we have: ←− ×−1 ←− ×−1 Sing −•(0) = Sing • (0) × Sg • , Sing −•(1 − z) = Sing • (z) .

1 1 Sketch of proof: Use + + 1 = 0 . eX − 1 e−X − 1

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) . Reﬂexion Formula for Multiple Bernoulli Polynomials

Corollary: For all z ∈ C, we have ←− −• • • • Beeg0 (1 − z) × Beeg0 (z) = 1 + I , 1 if r = 0 . 1 if r = 1 . where 1X1,··· ,Xr = and I X1,··· ,Xr = 0 if r > 0 . 0 if r 6= 1 .

Theorem: For all z ∈ C, we have: ←− ←− • • Beeg −•(1 − z) × Beeg (z) = beeg −• × (1• + I •) × beeg . Outline

1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function

2 The Form of a Multiple Bernoulli Polynomial

3 The General Reﬂexion Formula of Multiple Bernoulli Polynomial

4 An Example of Multiple Bernoulli Polynomial A Mould Equation

←− −• Goal: Solve the mould equation X eeg • × (1• + I •) × X eeg = 1• .

• • • Consequence: Deﬁne Beeg = X eeg × Beeg0 , so that:

←− • Beeg −•(1 − z) × Beeg (z) = 1• .

Deﬁnition: √ The mould Sg • is deﬁned by:

p (−1)r 2r Sg X1,··· ,Xr = 22r r Resolution of the Mould Equation

Property: ←− −• Any solution X eeg • of X eeg • × (1• + I •) × X eeg = 1• comes from a mould Zeg • satisfying:

←− ( −• Zeg • + Zeg = 0• , Zeg • is “primitive” , √ and is given by: X eeg • = Exp(Zeg •) × Sg • .

Sketch of proof: group-like = Exp(primitive) . Deﬁnition of Multiple Bernoulli Polynomials and Numbers

The mould Zeg • deﬁned by

 ∅  Zeg := 0 , (−1)r−1 Zeg X1,··· ,Xr := Zeg X1+···+Xr ,  r 1 1 1 where Zeg X = − + , satiﬁes the required conditions: eX − 1 X 2

←− ( −• Zeg • + Zeg = 0 , Zeg • is “primitive” ,

Deﬁnition: The moulds Beeg •(z) and beeg • are deﬁned by: √ Beeg •(z) = Exp(Zeg •) × Sg • × (Sing •(0))×−1 × Sing •(z) . √ beeg • = Exp(Zeg •) × Sg •

The coeﬃcients Be•(z) and Be• of these exponential generating series are called respectively the Multiple Bernoulli Polynomials and Numbers. one out of four Multiple Bernoulli Numbers is null. one out of two diagonals is “constant”. cross product around the zeros are equals : 28800 · 127008 = 604802 . “symmetrie” relatively to p = q .

Table of Multiple Bernoulli Numbers in length 2

bep,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of two diagonals is “constant”. cross product around the zeros are equals : 28800 · 127008 = 604802 . “symmetrie” relatively to p = q .

Table of Multiple Bernoulli Numbers in length 2

bep,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of four Multiple Bernoulli Numbers is null. cross product around the zeros are equals : 28800 · 127008 = 604802 . “symmetrie” relatively to p = q .

Table of Multiple Bernoulli Numbers in length 2

bep,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of four Multiple Bernoulli Numbers is null. one out of two diagonals is “constant”. “symmetrie” relatively to p = q .

Table of Multiple Bernoulli Numbers in length 2

bep,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of four Multiple Bernoulli Numbers is null. one out of two diagonals is “constant”. cross product around the zeros are equals : 28800 · 127008 = 604802 . Table of Multiple Bernoulli Numbers in length 2

bep,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of four Multiple Bernoulli Numbers is null. one out of two diagonals is “constant”. cross product around the zeros are equals : 28800 · 127008 = 604802 . “symmetrie” relatively to p = q . Construction Table of Multiple Bernoulli Numbers in length 2