Generalization of Bernoulli numbers and polynomials to the multiple case Olivier Bouillot, Marne-la-Vall´ ee University, France C.A.L.I.N. team seminary. Tuesday, 3 th March 2015 .
Introduction Definition: The numbers Z e s 1 , ··· , s r defined by � 1 Z e s 1 , ··· , s r = n 1 s 1 · · · n r s r , 0 < n r < ··· < n 1 where s 1 , · · · , s r ∈ C such that ℜ ( s 1 + · · · + s k ) > k , k ∈ [ [ 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 FKMT (0) = 1 Z e 0 , − 2 Z e 0 , − 2 Z e 0 , − 2 MP (0) = , (0) = , 18 . GZ 720 120 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: � � b n t n te xt B n ( x ) t n t e t − 1 = e t − 1 = n ! . n ! . n ≥ 0 n ≥ 0 By a recursive formula: By a recursive formula: B 0 ( x ) = 1 , b 0 = 1 , � n + 1 � ∀ n ∈ N , B ′ n +1 ( x ) = ( n + 1) B n ( x ) , � n � 1 ∀ n ∈ N , b k = 0 . ∀ n ∈ N ∗ , k B n ( x ) dx = 0 . k =0 0 First examples: First examples: b n = 1 , − 1 2 , 1 6 , 0 , − 1 30 , 0 , 1 42 , · · · B 0 ( x ) = 1 , x − 1 B 1 ( x ) = 2 , x 2 − x + 1 B 2 ( x ) = 6 , . . .
Elementary properties satisfied by the Bernoulli polynomials and numbers b 2 n +1 = 0 if n > 0 . P1 P2 B n (0) = B n (1) if n > 1 . � m + 1 � m � P3 b k = 0 , m > 0 . k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . � n � � n P4 B k ( x ) y n − k for all n . B n ( x + y ) = k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 N − 1 � k n = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 for all m > 0 and n ≥ 0. P9 B n m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . � n � n � P4 B k ( x ) y n − k for all n . B n ( x + y ) = k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . � n � n � P4 B k ( x ) y n − k for all n . B n ( x + y ) = k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . Important property, but turns � n � n out to have a generalization � P4 B k ( x ) y n − k for all n . B n ( x + y ) = with a corrective term... k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . Important property, but turns � n � n out to have a generalization � P4 B k ( x ) y n − k for all n . B n ( x + y ) = with a corrective term... k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 Has to be extended, but how??? ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . Important property, but turns � n � n out to have a generalization � P4 B k ( x ) y n − k for all n . B n ( x + y ) = with a corrective term... k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 Has to be extended, but how??? ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 Has to be extended, but how??? N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . Important property, but turns � n � n out to have a generalization � P4 B k ( x ) y n − k for all n . B n ( x + y ) = with a corrective term... k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 Has to be extended, but how??? ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 Has to be extended, but how??? N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . Does not depend of the Bernoulli numbers... n + 1 k =0 � x B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . n + 1 a � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
Elementary properties satisfied by the Bernoulli polynomials and numbers P1 b 2 n +1 = 0 if n > 0 . Have to be extended, B n (0) = B n (1) if n > 1 . but is not restritive enough . P2 � m + 1 � � m Has to be extended, but too particular. b k = 0 , m > 0 . P3 k k =0 B ′ n ( z ) = nB n − 1 ( z ) if n > 0 . Important property, but turns � n � n out to have a generalization � P4 B k ( x ) y n − k for all n . B n ( x + y ) = with a corrective term... k k =0 B n ( x + 1) − B n ( x ) = nx n − 1 , for all n . P5 Has to be extended, but how??? ( − 1) n B n (1 − x ) = B n ( x ), for all n . P6 Has to be extended, but how??? N − 1 � k p = B n +1 ( N ) − B n +1 (0) P7 . Does not depend of the Bernoulli numbers... n + 1 k =0 � x Has a generalization using the derivative B n ( t ) dt = B n +1 ( x ) − B n +1 ( a ) P8 . of a multiple Bernoulli polynomial instead n + 1 a of the Bernoulli polynmials. � � m − 1 � x + k B n ( mx ) = m n − 1 P9 B n for all m > 0 and n ≥ 0. m k =0
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