Multiple Bernoulli Polynomials and Numbers

Multiple Bernoulli Polynomials and Numbers

Olivier Bouillot, Paris-Sud University

Abstract

The aim of this work is to describe what can be multiple Bernoulli polynomials. In order to do this, we solve a system of difference equations generalizing the classical difference equation satisfied by the Bernoulli polynomials.

We also require that these polynomials span an algebra whose product is given by same rule as theM basis of QSym. Although there is not a unique solution, we construct an explicit and interesting solution, thanks to the reflexion equation satisfied by the Bernoulli polynomials.

R´esum´e

L’objectif de ce travail est de d´ecrire ce que peuvent ˆetre des polynˆomes de Bernoulli multiples. Pour cela, nous r´esolvons un syst`eme d’´equations aux diff´erences g´en´eralisant la classique ´equation aux diff´erences v´erifi´ee par les polynˆomes de Bernoulli. Nous imposons aussi que ces polynˆomes engendrent une alg`ebre dont le produit est celui de la baseM deQSym. Bien qu’il n’y ait pas unicit´e de la notion, l’´etude de l’´equation de reflexion v´erifi´ee par les polynˆomes de Bernoulli classique permet de construire une solution explicite et int´eressante combinatoirement.

Key words: Bernoulli polynomials and numbers, Quasi-symmetric functions, Generating series, Mould calculus.

The aim of this paper is to define a family of polynomials B^n1,···,nr, r ∈ N^∗, depending on non- negative integersn₁,· · · , n_r that generalizes the classical family of Bernoulli polynomials Bn+1

n+ 1, n≥0.

They will be calledmultiple divided) Bernoulli polynomials. In association to these polynomials, we will define multiple Bernoulli numbers, denoted byb^n1,···,nr, as the constant terms of the multiple Bernoulli polynomials.

1. Required conditions on multiple Bernoulli polynomials

It is well-known that the Riemann zeta function s7−→ ζ_R(s) (resp. the Hurwitz zeta function s7−→

ζH(s, z)) have a meromorphic continuation toCwith a unique pole at 1, whose values on negative integers are related to the Bernoulli numbers (resp. the Bernoulli polynomials):

ζ_R(−s) = (−1)^sBs+1

s+ 1 , ζ_H(−s, z) = (−1)^sBs+1(z)

s+ 1 (1)

These two functions have some well-known extension to the multiple case, respectively called multiple zeta values (MZV for short) and Hurwitz multiple zeta function, wheres1,· · · , sr∈Csuch that<(s1+

· · ·+sk)> k,k∈[[ 1 ;r]] (See [19], [21] for the MZV; see [2], [3] for the Hurwitz multiple zeta functions):

Ze^s1,···,sr = X

0<n_r<···<n1

1

n1s1· · ·nrsr , He^s1,···,sr :z7−→ X

0<n_r<···<n1

1

(n1+z)^s1· · ·(nr+z)^sr (2)

Email address: olivier.bouillot@villebon-charpak.fr(Olivier Bouillot, Paris-Sud University ).

URL:http://www-igm.univ-mlv.fr/∼bouillot/(Olivier Bouillot, Paris-Sud University ).

1 . Partially supported by A.N.R. project C.A.R.M.A. (A.N.R.-12-BS01-0017)

It has been observed that multizetas values have a meromorphic continuation toC(seee.g., [8], [20]).

Thus, we can consider that their values on non-positive integers are (divided) multiple Bernoulli numbers, but there is not unicity of the notion: almost all non-positive integers are singularities of the meromorphic continuation of multizetas, which means that they are points of indeterminacy. For example:

from [10],Ze^0,−2_{F KM T} = 1

18 , from [12],Ze^0,−2_GZ = 1

120 , from [15],Ze^0,−2_{M P} = 7

720 . (3)

For the same reason, Hurwitz multiple zeta functions have a meromorphic continuation to C, which explain why multiple Bernoulli polynomials can be seen as the evaluation of an Hurwitz multiple zeta function on non positive integers. Thus, Hurwitz multizeta functions are a good guide to understand what can be the multiple Bernoulli polynomials. But once again, there is no unicity of the notion.

On the first hand, let us note that multiple zeta values and Hurwitz multiple zeta functions are a specialization of the basisM of monomial quasi-symmetric functions, withxn=n⁻¹andxn = (n+z)⁻¹, the order being · · · < 3 < 2 < 1. Therefore, multiple Hurwitz zeta function and multiple zeta values multiply by the product of theM’s, namely the stuffle product which is denoted by . It is recursively defined on words, and then extended by linearity to non-commutative polynomials or series over an alphabet Ω, which is assumed to have a commutative semi-group structure, denoted by +:







ε u =u ε = u .

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

(4) Consequently, multiple Bernoulli polynomial have to multiply by the stuffle product.

On the other hand, the Hurwitz multiple zeta function satisfy a nice difference equation:

He^s1,···,sr(z−1)− He^s1,···,sr(z) =





 1

z^sr ifr= 1 .

He^{s1,···,sr−1}(z)· 1

z^sr ifr≥2 .

(5) Once reinterpreted with negative integers and adapted to the difference equation satisfied by Bernoulli polynomials, the Hurwitz multiple zeta functions suggest us to base our study of multiple Bernoulli polynomials such that they satisfy an analogue of the difference equation (5) and multiply by the stuffle product, i.e. to satisfy the following system of recursively defined polynomials:















Bⁿ(z) = Bn+1(z)

n+ 1 , wheren≥0 ,

B^n1,···,nr(z+ 1)−B^n1,···,nr(z) =B^{n1,···,nr−1}(z)z^nr , wherer≥1 and n1,· · ·, nr≥0 , theB^n1,···,nr multiply by the stuffle product.

(6)

2. Transcription of the system in an algebraic way

Let us first consider the family of exponential generating functions (B^Y1,···,Yr), with r ∈ N and Y1,· · ·, Yr ∈ X, whose coefficients are polynomials B^n1,···,nr of C[z], over the infinite (commutative) alphabet X ={X1, X2, X3,· · · } of indeterminates. Then, in order to see it as a unique object, we will

interpret these generating series as the coefficients of a noncommutative series, over an infinite (noncom- mutative) alphabetA={a1, a2,· · · }.

B^Y1,···,Yr(z) = X

n1,···,nr≥0

B^n1,···,nr(z)Y₁ⁿ¹

n1! · · ·Y_r^nr

nr!, for allr∈N^∗ , Y1,· · ·, Yr∈X. (7) B(z) = 1 +X

r>0

X

k1,···,kr>0

B^{Xk1,···,Xkr}(z)ak₁· · ·ak_r∈C[[X]]hhAii (8) This will lead us with an object satisfying the multiplicative difference equation:

B(z+ 1) =B(z)· 1 +X

k>0

e^zXka_k

(9) We are asking that the polynomials B^n1,···,nr multiply the stuffle product. It has been shown in [4]

that the B^Y1,···,Yr also multiply the stuffle product. this has also been previously suggested by [6], [16]

or [18]. Consequently, it turns out that the seriesBis a group-like element ofC[z][[X]]hhAiiif we consider that the lettera∈Aare primitive. Finally, System (6) can be rewritten as















B(z+ 1) =B(z)·E(z) , where E(z) = 1 +X

k>0

e^zXka_k , Bis a group-like element ofC[z][[X]]hhAii,

hB(z)|aki= e^zXk e^Xk−1 − 1

Xk

.

(10)

Let us emphazise that such a construction is not so common, since the first idea is to consider the non-commutative series whose coefficients would have been the multiple Bernoulli polynomials. Such an idea is coming from the secondary mould symmetries, especially this calledsymmetril, from themould calculus developped by Jean Ecalle (see [9] or [4], as well as [2], [7] or [17] for a crash course on this topic) Consequently, to be more familiarized with such objects, let us have a look at the analogue M of B(z) where the multiple Bernoulli polynomials are replaced with the monomial functionsMI(x) ofQSym defined for a compositionI= (i₁,· · · , i_r) by (see [11], or [13] and [14] for a more recent presentation):

Mi1,···,ir(x) = X

0<n₁<···<nr

xⁱ_n¹₁· · ·xⁱ_n^r_r (11)

M^Y1,···,Yr(x) = X

n₁,···,n_r≥0

M_{n1+1,···,nr+1}(x)Y₁ⁿ¹

n₁! · · ·Y_r^nr

n_r! , for allY₁,· · ·, Y_r∈X (12) M= 1 +X

r>0

X

k1,···,kr>0

M^{Xk1,···,Xkr}(x)ak₁· · ·ak_r =

−→

Y

n>0

1 +X

k>0

xne^xnXkak

∈C[[x]][[X]]hhAii (13)

We can see here a natural factorization, which can be particularized for Hurwitz multiple zeta function or multiple zeta values byxn= (n+z)⁻¹ orxn=n⁻¹.

3. Description of the set of solutions

In one hand, the multiplicative difference equation (9) produces the natural seriesS(z) defined by

S(z) =

←−

Y

n>0

E(z−n) = 1 +X

r>0

X

k₁,···,kr>0

e^{z(Xk1+···Xkr)}

r

Y

i=1

(e^{Xk1+···+Xki}−1)

a_k1· · ·a_kr (14)

satisfies the difference equation from System (10), is group-like as a product of group-like elements, but is an element ofS(z)∈C[z]((X))hhAii. It turns out that we have produced a false solution of System (10).

Nevertheless, it is actually a series of first importance to define the multiple Bernoulli polynomials. Let us note that if it was not valued in the formal Laurent series, the seriesS would have been the best choice of the generating series of multiple Bernoulli polynomials.

On the other hand, the difference equation (comming from (6) and (7)) satisfied by the seriesB^X1,···,Xr defines recursively the generating series B^X1,···,Xr, r > 0, up to a constant, because ker ∆∩zC[z] = {0}. Consequently, there exists a unique family

B^X₀^1,···,Xr(z)

r≥0 of formal power series satisfying this difference equation vanishing at 0. It produces a seriesB0∈C[z][[X]]hhAiidefined by:

B0(z) = 1 +X

r>0

X

k₁,···,k_r>0

B^X₀^{k1,···,Xkr}(z)ak₁· · ·ak_r . (15) Let us emphasize that we have a surprisingly simple expression ofB₀:

Lemma 3.1. (i) The noncommutative seriesB₀ is a group-like element ofC[z][[X]]hhAii.

(ii) The seriesB₀ can be expressed in terms ofS:B₀(z) = S(0)−1

· S(z).

Consequently, the idea is now to find a correction of S, like B₀ which is an element of C[z][[X]]hhAii.

This is done by the following characterization of the solutions of System (10):

Proposition 3.2. Any familly of polynomials which are solution of (6) comes from a noncommutative seriesB∈C[z][[X]]hhAii such that there existsb∈C[[X]]hhAiisatisfying:

1.hb|Aki= 1

e^Xk−1 − 1

X_k 2. bis group-like 3. B=b· S(0)⁻¹

· S(z).

Therefore, we deduce the following theorem which algebraically explain the different values of (3).

Theorem 3.3. The subgroup of group-like series ofC[z][[X]]hhAii, with vanishing coefficients in length1, acts on the set of all possible multiple Bernoulli polynomials,i.e.the set of solutions of (10).

4. Generalization of the reflection equation

In order to generalize the reflection formula satisfied by the Bernoulli polynomials, we shall see how this property generalizes to B₀(z) according to Proposition 3.2. This step allows us to find some restrictive condition to define a nice example of a generalization of Bernoulli polynomials.

Let us begin by defining some suitable notations for the sequel. For generic series ofC[z][[XXX]]hhAAAii s(z) =X

r∈N

X

k1,···,kr>0

s^{Xk1,···,Xkr}(z)ak1· · ·akr , (16) we will respectively denote bys(z) andes(z) the reverse and retrograde series ofs(z):

s(z) =X

r∈N

X

k₁,···,k_r>0

s^{Xkr,···,Xk1}(z)a_k1· · ·a_kr . (17)

s(z) =e X

r∈N

X

k1,···,kr>0

s^{−Xk1,···,−Xkr}(z)ak₁· · ·ak_r . (18) Proposition 4.1. Let sg= 1 +X

r>0

X

k₁,···,k_r>0

(−1)^ra_k1· · ·a_kr = 1 +X

n>0

a_n−1

. Then,

1. S(0) =e S(0)⁻¹

·sg and S(1e −z) = S(z)⁻¹

. (19)

2. ∀z∈C, sg·Be0(1−z) = B0(z)−1

. (20)

Thanks to the previous property and given a group-like seriesb as in Proposition 3.2, we get that a multi-Bernoulli polynomial satisfies:

B(1e −z)·B(z) =eb·sg⁻¹·b. (21) Even if there is not unicity of the multi-Bernoulli polynomials, we would be interested in having a nice combinatorial candidate, for example based on a simple formula for its reflection equation,i.e.eb·sg⁻¹·b must be a simple element of C[z][[XXX]]hhAAAii. As we have seen before, the series S(z) and S(0) can be considered as a nice guide to guess, here, which sense the word “simple” has. SinceS(0)e ·sg⁻¹· S(0) = 1 , this suggests the following heuristic:

Heuristic 1. A reasonable candidate for a multi-Bernoulli polynomial comes from the coefficients of a seriesB(z) =b·B₀(z) wherebsatisfies:

1.hb|aki= 1

e^Xk−1 − 1 Xk

2.bis group-like 3.eb·sg⁻¹·b= 1 .

5. An example of multiple Bernoulli polynomials and numbers

According to Heuristic 1, we need to solve the equation eu·sg⁻¹·u = 1 where u ∈ C[z][[X]]hhAii is group-like. The solutions are given by the following

Proposition 5.1. Any group-like elementu∈C[z][[X]]hhAiisatifyingeu·sg⁻¹·u= 1comes from a primitive seriesv∈C[z][[X]]hhAii satisfyingv+ev= 0, and is given by u=exp(v)·√

sg, where :

√sg= 1 +X

r>0

X

k1,···,kr>0

(−1)^r 2^2r



 2r

r



ak₁· · ·ak_r

We now have to determine a nice primitive series v∈C[z][[X]]hhAiisatisfyingv+ev= 0 We necessarily have:

hv|aki= 1

e^Xk−1 − 1 Xk

+1

2 :=f(Xk). (22)

Let us emphasize that the appearance of a term one-half is a wonderful thing and produces a really nice series: it surprisingly deletes the only term with an odd Bernoulli number and thus appears to be a natural correction of the series of divided Bernoulli numbers. Consequently, fora∈A, the serieshv|akiis an odd formal series in X_k ∈X. This is enough surprising and welcome that we want to generalize this property to all the coefficients ofvproducing the following

Heuristic 2. For our problem, a reasonable primitive seriesvmight satisfiesev=−vandv=v .

According to Heuristic 2, the coefficients ofvon words of length 2 are necessarily given by:

hv|a_k1a_k2i=−1

2f(X_k1+X_k2). (23)

This suggests to consider the primitive seriesvdefined by:

hv|a_k1· · ·a_kri= (−1)^r−1

r f(X_k1+· · ·+X_kr). (24)

Definition5.2. With the previous noncommutative seriesv, the seriesB(z) andbdefined by







B(z) = exp(v)·p

Sg·(S(0))⁻¹· S(z) b= exp(v)·p

Sg

(25) are noncommutative series of C[z][[X]]hhAii whose coefficients are respectively the exponential generating functions of multiple Bernoulli polynomialsand multiple Bernoulli numbers.

Everything is explicit, as the following example show:

Example 1. The exponential generating function of bi-Bernoulli polynomials is:

X

n1,n2≥0

B^n1,n2(z)Xⁿ¹ n1!

Yⁿ² n2! = −1

2f(X+Y) +1

2f(X)f(Y)−1

2f(X) +3 8 +f(X) e^zY −1

e^Y −1 −1 2

e^zY −1 e^Y −1 + e^z(X+Y)−1

(e^X−1)(e^X+Y −1)− e^zY −1 (e^X−1)(e^Y −1) .

(26)

Consequently, we obtain Table 1, as well as explicit expressions like, forn1, n2>0(ifn1= 0orn2= 0, the expression are not so simple, which turn out to be the propagation ofb16= 0...):

b^n1,n2 =1 2

bn₁+1bn₂+1

(n₁+ 1)(n₂+ 1) − bn₁+n₂+1

n₁+n₂+ 1

. (27)

Other tables are available at http://www-igm.univ-mlv.fr/∼bouillot/tables of multiple bernoulli.pdf.

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b^p,q p= 0 p= 1 p= 2 p= 3 p= 4 p= 5 p= 6 p= 7 p= 8

q= 0 3

8 −1

12 0 1

120 0 − 1

252 0 1

240 0

q= 1 −1 24

1 288

1 240 − 1

2880 − 1 504

1 6048

1

480 − 1

5760 − 1 264

q= 2 0 1

240 0 − 1

504 0 1

480 0 − 1

264 0

q= 3 1 240 − 1

2880 − 1 504

1 28800

1 480 − 1

60480 − 1 264

1 57600

691 65520

q= 4 0 − 1

504 0 1

480 0 − 1

264 0 691

65520 0

q= 5 − 1 504

1 6048

1 480 − 1

60480 − 1 264

1 127008

691 65520 − 1

120960 −1 24 Table 1

The first values of the bi-Bernoulli numbers

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