Multiple Bernoulli Polynomials and Numbers

Transcription

1 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 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. Résumé L objectif de ce travail est de décrire ce que peuvent être des polynômes de Bernoulli multiples. Pour cela, nous résolvons un système d équations aux différences généralisant la classique équation aux différences vérifiée par les polynômes de Bernoulli. Nous imposons aussi que ces polynômes engendrent une algèbre dont le produit est celui de la base M de QSym. Bien qu il n y ait pas unicité de la notion, l étude de l équation de reflexion vérifiée par les polynômes de Bernoulli classique permet de construire une solution explicite et intéressante 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 n,,nr, r N, depending on nonnegative integers n,, n r that generalizes the classical family of Bernoulli polynomials B n+ n +, n. They will be called multiple divided) Bernoulli polynomials. In association to these polynomials, we will define multiple Bernoulli numbers, denoted by b n,,nr, as the constant terms of the multiple Bernoulli polynomials.. Required conditions on multiple Bernoulli polynomials It is well-known that the Riemann zeta function s ζ R s) resp. the Hurwitz zeta function s ζ H s, z)) have a meromorphic continuation to C with a unique pole at, whose values on negative integers are related to the Bernoulli numbers resp. the Bernoulli polynomials): ζ R s) = ) s B s+, ζ H s, z) = ) s B s+z) ) s + s + 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 s,, s r C such that Rs + + s k ) > k, k [ ; r ] See [9], [2] for the MZV; see [2], [3] for the Hurwitz multiple zeta functions): Ze s,,sr = n s <n nr s r, He s,,sr : z 2) n + z) s n r + z) sr r< <n <n r< <n address: olivier.bouillot@villebon-charpak.fr Olivier Bouillot, Paris-Sud University ). URL: bouillot/ Olivier Bouillot, Paris-Sud University ).. Partially supported by A.N.R. project C.A.R.M.A. A.N.R.-2-BS-7) Preprint submitted to the Compte Rendus de l Académie des Sciences April 2, 27

2 It has been observed that multizetas values have a meromorphic continuation to C see e.g., [8], [2]). 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 [], Ze, 2 F KMT = 8, from [2], Ze, 2 GZ = 2, from [5], Ze, 2 MP = ) 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 basis M of monomial quasi-symmetric functions, with x n = n and x n = n + z), the order being < 3 < 2 <. 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. 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: if r =. He s,,sr z ) He s,,sr z) = z sr He s, 5),sr z) z if r 2. sr 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: 4) B n z) = B n+z) n +, where n, B n,,nr z + ) B n,,nr z) = B n,,nr z)z nr, where r and n,, n r, n,,nr the B 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 Y,,Yr ), with r N and Y,, Y r X, whose coefficients are polynomials B n,,nr of C[z], over the infinite commutative) alphabet X = {X, X 2, X 3, } of indeterminates. Then, in order to see it as a unique object, we will 2

3 interpret these generating series as the coefficients of a noncommutative series, over an infinite noncommutative) alphabet A = {a, a 2, }. B Y,,Yr z) = n,,n r Bz) = + r> B n,,nr z) Y n n! Y r nr n r!, for all r N, Y,, Y r X. 7) B X k,,x kr z) ak a kr C[X] A 8) k,,k r> This will lead us with an object satisfying the multiplicative difference equation: Bz + ) = Bz) + ) e zx k a k k> 9) We are asking that the polynomials B n,,nr multiply the stuffle product. It has been shown in [4] that the B Y,,Yr also multiply the stuffle product. this has also been previously suggested by [6], [6] or [8]. Consequently, it turns out that the series B is a group-like element of C[z][X] A if we consider that the letter a A are primitive. Finally, System 6) can be rewritten as Bz + ) = Bz) Ez), where Ez) = B is a group-like element of C[z][X] A, Bz) a k = ezx k e X. k X k + ) e zx k a k, k> ) 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 [7] 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 Bz) where the multiple Bernoulli polynomials are replaced with the monomial functions M I x) of QSym defined for a composition I = i,, i r ) by see [], or [3] and [4] for a more recent presentation): M i,,i r x) = x i n x ir n r ) <n < <n r M Y,,Yr x) = M Y n n+,,n r+x) n! Y r nr, for all Y,, Y r X 2) n r! n,,n r M = + r> k,,k r> M X k,,x kr x) ak a kr = n> + x n e xnx k a k ) C[x][[X] A 3) k> We can see here a natural factorization, which can be particularized for Hurwitz multiple zeta function or multiple zeta values by x n = n + z) or x n = n. 3. Description of the set of solutions In one hand, the multiplicative difference equation 9) produces the natural series Sz) defined by 3

4 Sz) = n> Ez n) = + r> k,,k r> e zx k + X kr ) a r k a kr 4) e X k + +X ki ) satisfies the difference equation from System ), is group-like as a product of group-like elements, but is an element of Sz) C[z]X)) A. It turns out that we have produced a false solution of System ). 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. X,,Xr On the other hand, the difference equation comming from 6) and 7)) satisfied by the series B defines recursively the generating series B X,,Xr, r >, up) to a constant, because ker zc[z] = X,,Xr {}. Consequently, there exists a unique family B z) of formal power series satisfying this difference equation vanishing at. It produces a series B C[z][X] A defined by: B z) = + B X k,,x kr z) a k a kr. 5) r> k,,k r> Let us emphasize that we have a surprisingly simple expression of B : Lemma 3.. i) The noncommutative series B is a group-like element of C[z][X] A. ii) The series B can be expressed in terms of S: B z) = S) ) Sz). Consequently, the idea is now to find a correction of S, like B which is an element of C[z][X] A. This is done by the following characterization of the solutions of System ): Proposition 3.2. Any familly of polynomials which are solution of 6) comes from a noncommutative series B C[z][X] A such that there exists b C[X] A satisfying:. b A k = e X 2. b is group-like 3. B = b S) ) Sz). k X k 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] A, with vanishing coefficients in length, acts on the set of all possible multiple Bernoulli polynomials, i.e. the set of solutions of ). i= r 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 B 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 ] A sz) = s X k,,x kr z) ak a kr, 6) r N k,,k r> we will respectively denote by sz) and sz) the reverse and retrograde series of sz): sz) = s X kr,,x k z) ak a kr. 7) r N k,,k r> 4

5 sz) = r N Proposition 4.. Let sg = + r> k,,k r> k,,k r> s X k,, X kr z) ak a kr. 8) ) r a k a kr = + n> a n ). Then,. S) = S) ) sg and S z) = Sz) ). 9) 2. z C, sg B z) = B z) ). 2) 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: B z) Bz) = b sg b. 2) 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. b sg b must be a simple element of C[z][X ] A. As we have seen before, the series Sz) and S) can be considered as a nice guide to guess, here, which sense the word simple has. Since S) sg S) =, this suggests the following heuristic: Heuristic. A reasonable candidate for a multi-bernoulli polynomial comes from the coefficients of a series Bz) = b B z) where b satisfies:. b a k = e X k X k 2. b is group-like 3. b sg b =. 5. An example of multiple Bernoulli polynomials and numbers According to Heuristic, we need to solve the equation ũ sg u = where u C[z][X] A is group-like. The solutions are given by the following Proposition 5.. Any group-like element u C[z][X] A satifying ũ sg u = comes from a primitive series v C[z][X] A satisfying v + ṽ =, and is given by u = expv) sg, where : ) r sg = + 2r 2 2r a k a kr r r> k,,k r> We now have to determine a nice primitive series v C[z][X] A satisfying v + ṽ = We necessarily have: v a k = e X + k X k 2 := fx k). 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, for a A, the series v a k is 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 of v producing the following Heuristic 2. For our problem, a reasonable primitive series v might satisfies ṽ = v and v = v. 5

6 According to Heuristic 2, the coefficients of v on words of length 2 are necessarily given by: This suggests to consider the primitive series v defined by: v a k a k2 = 2 fx k + X k2 ). 23) v a k a kr = )r fx k + + X kr ). 24) r Definition 5.2. With the previous noncommutative series v, the series Bz) and b defined by Bz) = expv) Sg S)) Sz) b = expv) Sg 25) are noncommutative series of C[z][X] A 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. The exponential generating function of bi-bernoulli polynomials is: n,n 2 B n,n2 z) Xn n! Y n2 n 2! = 2 fx + Y ) + 2 fx)fy ) 2 fx) fx) ezy e Y 2 e zy e Y e zx+y ) + e X )e X+Y ) e zy e X )e Y ). Consequently, we obtain Table, as well as explicit expressions like, for n, n 2 > if n = or n 2 =, the expression are not so simple, which turn out to be the propagation of b...): b n,n2 = bn+b n2+ 2 n + )n 2 + ) b ) n +n ) n + n 2 + Other tables are available at bouillot/tables of multiple bernoulli.pdf. 26) References [] S. Akiyama, S. Egamiand, Y. Tanigawa : Analytic continuation of multiple zeta functions and their values at nonpositive integers, Acta Arithmetica, 98 2), n 2, p7-6. [2] O. Bouillot : The Algebra of Multitangent Functions, Journal of Algebra, 4 24), p [3] O. Bouillot : The Algebra of Hurwitz Multizeta Functions, C. R. Acad. Sci. Paris, Ser I 24), 6p. [4] O. Bouillot : Mould calculus On the secondary symmetries, C. R. Acad. Sci. Paris, Ser I 26), 6p. [5] P. Cartier : Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Astérisque ), Sém. Bourbaki n 5, p [6] F. Chapoton, F. Hivert, J.-C. Novelli, J.-Y. Thibon : An operational calculus for the Mould operad, Internat. Math. Research Notices IMRN 28), n 9:Art. ID rnn8, 22 pp. [7] J. Cresson : Calcul moulien, Annales de la faculté des sciences de Toulouse, Sér. 6 29), 8, n 2, p [8] J. Ecalle : ARI/GARI, la dimorphie et l arithmétique des multizêtas, un premier bilan, Journal de Théorie des Nombres de Bordeaux, 5 23), n 2, p

7 b p,q p = p = p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8 q = q = q = q = q = q = Table The first values of the bi-bernoulli numbers [9] J. Ecalle : Singularités non abordables par la géométrie, Annales de l institut Fourier, ), n 2, p [] 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: [] I. Gessel : Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra Boulder, Colo., 983), , Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 984. [2] L. Guo, B. Zhang : Renormalization of multiple zeta values, J. Algebra, 39 28), n 9, p [3] F. Hivert : Local Action of the Symmetric Group and Generalizations of Quasi-symmetric Functions, in 7th Annual International Conference on Formal Power Series and Algebraic Combinatorics, Taormina, 25. [4] C. Malvenuto, C. Reutenauer : Duality Between Quasi-symmetric Functions and the Solomon Descent Algebra, Journal of Algebra, n 77, 995, p [5] D. Manchon, S. Paycha : Nested sums of symbols and renormalised multiple zeta values, Int. Math. Res. Notices, 24, 2), p [6] F. Menous, J.-C. Novelli, J.-Y. Thibon : Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras, Advances in Applied Math., 5, n 2, 23), p [7] 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, 5, European Mathematical Society, Zürich, 29, pp [8] J.-Y. Thibon : Noncommutative symmetric functions and combinatorial Hopf algebras, in Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation, Vol, O.Costin, F.Fauvet, F.Menous, D.Sauzin. Eds., Ann. Scuo.Norm.Pisa, 2), Vol. 2, p [9] M. Waldschmidt : Valeurs zêta multiples. Une introduction, Journal de Théorie des Nombres de Bordeaux, 2 2), p [2] J. Zhao : Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 28, 2), p [2] W. Zudilin : Algebraic relations for multiple zeta values, Russian Mathematical Surveys, 58, 23, p