Multiple Bernoulli Polynomials and Numbers Olivier Bouillot, Paris-Sud University 1
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 the M 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´ecrirece que peuvent ˆetredes polynˆomesde Bernoulli multiples. Pour cela, nous r´esolvons un syst`emed’´equationsaux diff´erencesg´en´eralisant la classique ´equationaux diff´erencesv´erifi´eepar les polynˆomesde Bernoulli. Nous imposons aussi que ces polynˆomesengendrent une alg`ebredont le produit est celui de la base M de QSym. Bien qu’il n’y ait pas unicit´ede la notion, l’´etudede l’´equationde reflexion v´erifi´eepar les polynˆomesde 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 Bn1,··· ,nr , r ∈ N∗, depending on non- B negative integers n , ··· , n that generalizes the classical family of Bernoulli polynomials n+1 , n ≥ 0. 1 r n + 1 They will be called multiple divided) Bernoulli polynomials. In association to these polynomials, we will define multiple Bernoulli numbers, denoted by bn1,··· ,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 s 7−→ ζR(s) (resp. the Hurwitz zeta function s 7−→ ζH(s, z)) have a meromorphic continuation to C with a unique pole at 1, whose values on negative integers are related to the Bernoulli numbers (resp. the Bernoulli polynomials): B B (z) ζ (−s) = (−1)s s+1 , ζ (−s, z) = (−1)s s+1 (1) R s + 1 H s + 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, where s1, ··· , sr ∈ C such that <(s1 + ··· + sk) > k, k ∈ [[1 ; r ]] (See [19], [21] for the MZV; see [2], [3] for the Hurwitz multiple zeta functions): 1 1 s1,··· ,sr X s1,··· ,sr X Ze = s s , He : z 7−→ s s (2) n1 1 ··· nr r (n1 + z) 1 ··· (nr + z) r 0 Email address: [email protected] (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) Preprint submitted to the Compte Rendus de l’Acad´emie des Sciences April 21, 2017 It has been observed that multizetas values have a meromorphic continuation to C (see e.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: 1 1 7 from [10], Ze0,−2 = , from [12], Ze0,−2 = , from [15], Ze0,−2 = . (3) FKMT 18 GZ 120 MP 720 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 −1 −1 specialization of the basis M of monomial quasi-symmetric functions, with xn = n and xn = (n + z) , the order being ··· < 3 < 2 < 1. Therefore, multiple Hurwitz zeta function and multiple zeta values multiply by the product of the M’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 . (4) ua vb = (u vb) a + (ua v) b + (u v)(a + b) . 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: 1 s if r = 1 . Hes1,··· ,sr (z − 1) − Hes1,··· ,sr (z) = z r (5) s ,··· ,s 1 He 1 r−1 (z) · if r ≥ 2 . zsr 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: n Bn+1(z) B (z) = , where n ≥ 0 , n + 1 n1,··· ,nr n1,··· ,nr n1,··· ,nr−1 nr (6) B (z + 1) − B (z) = B (z)z , where r ≥ 1 and n1, ··· , nr ≥ 0 , the Bn1,··· ,nr multiply by the stuffle product. 2. Transcription of the system in an algebraic way Let us first consider the family of exponential generating functions (BY1,··· ,Yr ), with r ∈ N and n1,··· ,nr Y1, ··· ,Yr ∈ X, whose coefficients are polynomials B 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 2 interpret these generating series as the coefficients of a noncommutative series, over an infinite (noncom- mutative) alphabet A = {a1, a2, ···}. Y n1 Y nr Y1,··· ,Yr X n1,··· ,nr 1 r ∗ B (z) = B (z) ··· , for all r ∈ N ,Y1, ··· ,Yr ∈ X . (7) n1! nr! n1,··· ,nr ≥0 X X Xk1 ,··· ,Xkr B(z) = 1 + B (z) ak1 ··· akr ∈C[[X]]hhAii (8) r>0 k1,··· ,kr >0 This will lead us with an object satisfying the multiplicative difference equation: X zXk B(z + 1) = B(z) · 1 + e ak (9) k>0 We are asking that the polynomials Bn1,··· ,nr multiply the stuffle product. It has been shown in [4] that the BY1,··· ,Yr also multiply the stuffle product. this has also been previously suggested by [6], [16] or [18]. Consequently, it turns out that the series B is a group-like element of C[z][[X]]hhAii if we consider that the letter a ∈ A are primitive. Finally, System (6) can be rewritten as X B(z + 1) = B(z) · E(z) , where E(z) = 1 + ezXk a , k k>0 (10) B is a group-like element of C[z][[X]]hhAii , ezXk 1 hB(z)|aki = X − . e k − 1 Xk 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 called symmetril, from the mould 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 functions MI (x) of QSym defined for a composition I = (i1, ··· , ir) by (see [11], or [13] and [14] for a more recent presentation): X M (x) = xi1 ··· xir (11) i1,··· ,ir n1 nr 0 3. Description of the set of solutions In one hand, the multiplicative difference equation (9) produces the natural series S(z) defined by 3 ←− z(X +···X ) Y X X e k1 kr S(z) = E(z − n) = 1 + r ak1 ··· akr (14) n>0 r>0 k ,··· ,k >0 Y Xk +···+Xk 1 r (e 1 i − 1) i=1 satisfies the difference equation from System (10), is group-like as a product of group-like elements, but is an element of S(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 series S 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 series BX1,··· ,Xr defines recursively the generating series BX1,··· ,Xr , r > 0, up to a constant, because ker ∆ ∩ zC[z] = X1,··· ,Xr {0}. Consequently, there exists a unique family B0 (z) of formal power series satisfying this r≥0 difference equation vanishing at 0. It produces a series B0 ∈ C[z][[X]]hhAii defined by: X X Xk1 ,··· ,Xkr B0(z) = 1 + B0 (z) ak1 ··· akr . (15) r>0 k1,··· ,kr >0 Let us emphasize that we have a surprisingly simple expression of B0: Lemma 3.1. (i) The noncommutative series B0 is a group-like element of C[z][[X]]hhAii. −1 (ii) The series B0 can be expressed in terms of S: B0(z) = S(0) ·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 series B ∈ C[z][[X]]hhAii such that there exists b ∈ C[[X]]hhAii satisfying: 1 1 −1 1. hb|Aki = X − 2. b is group-like 3. B = b · S(0) ·S(z) . e k − 1 Xk Therefore, we deduce the following theorem which algebraically explain the different values of (3). Theorem 3.3. The subgroup of group-like series of C[z][[X]]hhAii, with vanishing coefficients in length 1, 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 of C[z][[X]]hhAii X X Xk1 ,··· ,Xkr s(z) = s (z) ak1 ··· akr , (16) r∈N k1,··· ,kr >0 we will respectively denote by s(z) and se(z) the reverse and retrograde series of s(z): X X Xkr ,··· ,Xk1 s(z) = s (z) ak1 ··· akr . (17) r∈N k1,··· ,kr >0 4 X X −X ,··· ,−X s(z) = s k1 kr (z) a ··· a . (18) e k1 kr r∈N k1,··· ,kr >0 −1 X X r X Proposition 4.1. Let sg = 1 + (−1) ak1 ··· akr = 1 + an . Then, r>0 k1,··· ,kr >0 n>0 −1 −1 1. Se(0) = S(0) · sg and Se(1 − z) = S(z) . (19) −1 2. ∀z ∈ C , sg · Be 0(1 − z) = B0(z) . (20) Thanks to the previous property and given a group-like series b as in Proposition 3.2, we get that a multi-Bernoulli polynomial satisfies: Be (1 − 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][[X]]hhAii. 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. Since Se(0) · 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 series B(z) = b · B0(z) where b satisfies: 1 1 −1 1. hb|aki = X − 2. b is group-like 3. eb · sg · b = 1 . e k − 1 Xk 5. An example of multiple Bernoulli polynomials and numbers −1 According to Heuristic 1, we need to solve the equation eu · sg · u = 1 where u ∈ C[z][[X]]hhAii is group-like. The solutions are given by the following Proposition 5.1. Any group-like element u ∈ [z][[X]]hhAii satifying u·sg−1 ·u = 1 comes from a primitive C e √ series v ∈ C[z][[X]]hhAii satisfying v + ev = 0, and is given by u = exp(v) · sg, where : √ X X (−1)r 2r sg = 1 + a ··· a 22r k1 kr r>0 k1,··· ,kr >0 r We now have to determine a nice primitive series v ∈ C[z][[X]]hhAii satisfying v + ev = 0 We necessarily have: 1 1 1 hv|aki = X − + := f(Xk) . (22) e k − 1 Xk 2 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, for a ∈ A, the series hv|aki is an odd formal series in Xk ∈ X. This is enough surprising and welcome that we want to generalize this property to all the coefficients of v producing the following Heuristic 2. For our problem, a reasonable primitive series v might satisfies ev = −v and v = v . 5 According to Heuristic 2, the coefficients of v on words of length 2 are necessarily given by: 1 hv|a a i = − f(X + X ) . (23) k1 k2 2 k1 k2 This suggests to consider the primitive series v defined by: (−1)r−1 hv|a ··· a i = f(X + ··· + X ) . (24) k1 kr r k1 kr Definition 5.2. With the previous noncommutative series v, the series B(z) and b defined by p −1 B(z) = exp(v) · Sg · (S(0)) ·S(z) (25) b = exp(v) · pSg are noncommutative series of C[z][[X]]hhAii whose coefficients are respectively the exponential generating functions of multiple Bernoulli polynomials and multiple Bernoulli numbers. Everything is explicit, as the following example show: Example 1. The exponential generating function of bi-Bernoulli polynomials is: X Xn1 Y n2 1 1 1 3 Bn1,n2 (z) = − f(X + Y ) + f(X)f(Y ) − f(X) + n1! n2! 2 2 2 8 n1,n2≥0 ezY − 1 1 ezY − 1 +f(X) − (26) eY − 1 2 eY − 1 ez(X+Y ) − 1 ezY − 1 + − . (eX − 1)(eX+Y − 1) (eX − 1)(eY − 1) Consequently, we obtain Table 1, as well as explicit expressions like, for n1, n2 > 0 (if n1 = 0 or n2 = 0, the expression are not so simple, which turn out to be the propagation of b1 6= 0...): 1 bn +1bn +1 bn +n +1 bn1,n2 = 1 2 − 1 2 . (27) 2 (n1 + 1)(n2 + 1) n1 + n2 + 1 Other tables are available at http://www-igm.univ-mlv.fr/∼bouillot/tables of multiple bernoulli.pdf. References [1] S. Akiyama, S. Egamiand, Y. Tanigawa : Analytic continuation of multiple zeta functions and their values at non- positive integers, Acta Arithmetica, 98 (2001), n◦2, p107-116. [2] O. Bouillot : The Algebra of Multitangent Functions, Journal of Algebra, 410 (2014), p. 148-238. [3] O. Bouillot : The Algebra of Hurwitz Multizeta Functions, C. R. Acad. Sci. Paris, Ser I (2014), 6p. [4] O. Bouillot : Mould calculus On the secondary symmetries, C. R. Acad. Sci. Paris, Ser I (2016), 6p. [5] P. Cartier : Fonctions polylogarithmes, nombres polyzˆetaset groupes pro-unipotents, Ast´erisque282 (2002), S´em. Bourbaki n◦5, p. 137-173. [6] F. Chapoton, F. Hivert, J.-C. Novelli, J.-Y. Thibon : An operational calculus for the Mould operad, Internat. Math. Research Notices IMRN (2008), n◦9:Art. ID rnn018, 22 pp. [7] J. Cresson : Calcul moulien, Annales de la facult´edes sciences de Toulouse, S´er.6 (2009), 18, n◦2, p. 307-395. [8] J. Ecalle : ARI/GARI, la dimorphie et l’arithm´etiquedes multizˆetas,un premier bilan, Journal de Th´eoriedes Nombres de Bordeaux, 15 (2003), n◦2, p. 411-478. 6 bp,q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8 3 1 1 1 1 q = 0 − 0 0 − 0 0 8 12 120 252 240 1 1 1 1 1 1 1 1 1 q = 1 − − − − − 24 288 240 2880 504 6048 480 5760 264 1 1 1 1 q = 2 0 0 − 0 0 − 0 240 504 480 264 1 1 1 1 1 1 1 1 691 q = 3 − − − − 240 2880 504 28800 480 60480 264 57600 65520 1 1 1 691 q = 4 0 − 0 0 − 0 0 504 480 264 65520 1 1 1 1 1 1 691 1 1 q = 5 − − − − − 504 6048 480 60480 264 127008 65520 120960 24 Table 1 The first values of the bi-Bernoulli numbers [9] J. Ecalle : Singularit´esnon abordables par la g´eom´etrie, Annales de l’institut Fourier, 42 (1992), n◦2, p. 73-164. [10] H. Furusho, Y.Komori, K.Matsumoto and H.Tsumura : Desingularization of Complex Multiple Zeta-Functions, Fundamentals of p-adic multiple L-functions, and evaluation of their special values, preprint ArXiv:1309.3982. [11] I. Gessel : Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), 289-317, Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984. [12] L. Guo, B. Zhang : Renormalization of multiple zeta values, J. Algebra, 319 (2008), n◦9, p. 3770-3809. [13] F. Hivert : Local Action of the Symmetric Group and Generalizations of Quasi-symmetric Functions, in 17th Annual International Conference on Formal Power Series and Algebraic Combinatorics, Taormina, 2005. [14] C. Malvenuto, C. Reutenauer : Duality Between Quasi-symmetric Functions and the Solomon Descent Algebra, Journal of Algebra, n◦177, 1995, p. 967-982. [15] D. Manchon, S. Paycha : Nested sums of symbols and renormalised multiple zeta values, Int. Math. Res. Notices, 24, (2010), p. 4628-4697. [16] F. Menous, J.-C. Novelli, J.-Y. Thibon : Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras, Advances in Applied Math., 51, n◦2, (2013), p. 177-227. [17] D. Sauzin : Mould Expansion for the Saddle-node and Resurgence Monomials, in “Renormalization and Galois theories”, A. Connes, F. Fauvet, J. P. Ramis. Eds., IRMA Lectures in Mathematics and Theoretical Physics, 15, European Mathematical Society, Z¨urich, 2009, pp. 83-163. [18] J.-Y. Thibon : Noncommutative symmetric functions and combinatorial Hopf algebras, in “Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation, Vol 1”, O.Costin, F.Fauvet, F.Menous, D.Sauzin. Eds., Ann. Scuo.Norm.Pisa , (2011), Vol. 12, p. 219-258. [19] M. Waldschmidt : Valeurs zˆeta multiples. Une introduction, Journal de Th´eoriedes Nombres de Bordeaux, 12 (2000), p. 581-592. [20] J. Zhao : Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 128, (2000), p. 1275-1283. [21] W. Zudilin : Algebraic relations for multiple zeta values, Russian Mathematical Surveys, 58, 2003, p. 1-29. 7