Generalization of Bernoulli numbers and polynomials to the multiple case Olivier Bouillot, Marne-la-Vall´eeUniversity, France C.A.L.I.N. team seminary. Tuesday, 3th March 2015 . Introduction Definition: The numbers Zes1;··· ;sr defined by s1;··· ;sr X 1 Ze = s s ; n1 1 ··· nr r 0<nr <···<n1 where s1; ··· ; sr 2 C such that <(s1 + ··· + sk ) > k, k 2 [[1 ; r ]], are called multiple zeta values. Fact: There exist at least three different way to renormalize multiple zeta values at negative integers. 7 1 1 Ze0;−2(0) = ; Ze0;−2(0) = ; Ze0;−2 (0) = : MP 720 GZ 120 FKMT 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. Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Hurwitz Zeta Function and Hurwitz multiple zeta functions Reminders on Quasi-Symmetric Functions 2 Algebraic reformulation of the problem 3 The Structure of a Multiple Bernoulli Polynomial 4 The General Reflexion Formula of Multiple Bernoulli Polynomial 5 An Example of Multiple Bernoulli Polynomial 6 An algorithm to compute the double Bernoulli Numbers 7 Properties satisfied by Bernoulli polynomials and numbers Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Hurwitz Zeta Function and Hurwitz multiple zeta functions Reminders on Quasi-Symmetric Functions 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: 8 8 b = 1 ; B0(x) = 1 ; > 0 > 0 < n < 8n 2 N ; Bn+1(x) = (n + 1)Bn(x) ; X n + 1 Z 1 8n 2 N ; bk = 0 : ∗ > k > 8n 2 ; Bn(x) dx = 0 : : k=0 :> N 0 First examples: First examples: 1 1 1 1 b = 1; − ; ; 0; − ; 0; ; ··· B (x) = 1 ; n 2 6 30 42 0 1 B (x) = x − ; 1 2 2 1 B2(x) = x − x + ; . 6 . Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b2n+1 = 0 if n > 0: P2 Bn(0) = Bn(1) if n > 1: m X m + 1 P3 b = 0; m > 0: k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 < n P4 X n n−k B (x + y) = B (x)y for all n: :> n k k k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. n P6 (−1) Bn(1 − x) = Bn(x), for all n. N−1 X n Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Has to be extended, but too particular. Important property, but turns out to have a generalization with a corrective term... Has to be extended, but how??? Has to be extended, but how??? Does not depend of the Bernoulli numbers... Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough. m X m + 1 P3 b = 0; m > 0: k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 < n P4 X n n−k B (x + y) = B (x)y for all n: :> n k k k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. n P6 (−1) Bn(1 − x) = Bn(x), for all n. N−1 X p Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Important property, but turns out to have a generalization with a corrective term... Has to be extended, but how??? Has to be extended, but how??? Does not depend of the Bernoulli numbers... Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough. m X m + 1 P3 b = 0; m > 0: Has to be extended, but too particular. k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 < n P4 X n n−k B (x + y) = B (x)y for all n: :> n k k k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. n P6 (−1) Bn(1 − x) = Bn(x), for all n. N−1 X p Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Has to be extended, but how??? Has to be extended, but how??? Does not depend of the Bernoulli numbers... Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough. m X m + 1 P3 b = 0; m > 0: Has to be extended, but too particular. k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 Important property, but turns < n out to have a generalization P4 X n n−k B (x + y) = B (x)y for all n: :> n k k with a corrective term... k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. n P6 (−1) Bn(1 − x) = Bn(x), for all n. N−1 X p Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Has to be extended, but how??? Does not depend of the Bernoulli numbers... Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough. m X m + 1 P3 b = 0; m > 0: Has to be extended, but too particular. k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 Important property, but turns < n out to have a generalization P4 X n n−k B (x + y) = B (x)y for all n: :> n k k with a corrective term... k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. Has to be extended, but how??? n P6 (−1) Bn(1 − x) = Bn(x), for all n. N−1 X p Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Does not depend of the Bernoulli numbers... Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough. m X m + 1 P3 b = 0; m > 0: Has to be extended, but too particular. k k k=0 8 B0 (z) = nB (z) if n > 0: > n n−1 Important property, but turns < n out to have a generalization P4 X n n−k B (x + y) = B (x)y for all n: :> n k k with a corrective term... k=0 n−1 P5 Bn(x + 1) − Bn(x) = nx , for all n. Has to be extended, but how??? n P6 (−1) Bn(1 − x) = Bn(x), for all n. Has to be extended, but how??? N−1 X p Bn+1(N) − Bn+1(0) P7 k = : n + 1 k=0 Z x Bn+1(x) − Bn+1(a) P8 Bn(t) dt = : a n + 1 m−1 n−1 X k P9 B (mx) = m B x + for all m > 0 and n ≥ 0. n n m k=0 Has a generalization using the derivative of a multiple Bernoulli polynomial instead of the Bernoulli polynmials. ??? Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b = 0 if n > 0: 9 2n+1 = Have to be extended, P2 Bn(0) = Bn(1) if n > 1: ; but is not restritive enough.