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Multiple Bernoulli Polynomials and Numbers

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Multiple Bernoulli Polynomials and Numbers Multiple Bernoulli Polynomials and Numbers Olivier Bouillot, Marnes-la-Vall´eeUniversity, France 72th Sminaire Lotharingien de Combinatoire, Lyon, Wenesday 26th March 2014 . Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function 2 The Form of a Multiple Bernoulli Polynomial 3 The General Reflexion Formula of Multiple Bernoulli Polynomial 4 An Example of Multiple Bernoulli Polynomial 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 > Bn(x) dx = 0 : : k=0 :> 0 First examples: First examples: 1 1 1 1 bn = 1; − ; ; 0; − ; 0; ; ··· B0(x) = 1 ; 2 6 30 42 1 B (x) = x − ; 1 2 1 B (x) = x 2 − x + ; 2 6 . Some Properties of Bernoulli Polynomials Property 1: Difference Equation n−1 ∗ ∆(Bn)(x) = nx for all n 2 N , where ∆(f )(z) = f (z + 1) − f (z). Property 2: Reflexion Formula n (−1) Bn(1 − x) = Bn(x) for all n 2 N . Property 3: Expression in the falling factorial basis n X n + 1 n B (x) − b = (x) for all n 2 , where: n+1 n+1 k + 1 k k+1 N k=0 • (x)n = x(x − 1) ··· (x − n + 1) . n • denotes the Stirling number of second kind. k On the Riemann Zeta Function and Hurwitz Zeta Function Definition: The Riemann Zeta Function and Hurwitz Zeta Function are defined, for <e s > 1, and z 2 C − N<0, by: X 1 X 1 ζ(s) = ; ζ(s; z) = : ns (n + z)s n≥0 n≥0 Property: s 7−! ζ(s) and s 7−! ζ(s; z) can be analytically extend to a meromorphic function on C, with a simple pole located at 1. b Remark: ζ(−n) = − n+1 for all n 2 : n + 1 N B (z) ζ(−n; z) = − n+1 for all n 2 : n + 1 N Multiple Zeta Values and Hurwitz Multiple Zeta Functions Definition of Hurwitz Multiple Zeta Functions s1;··· ;sr X 1 He (z) = s s ; if z 2 C − N<0 and (n1 + z) 1 ··· (nr + z) r 0<nr <···<n1 ∗ r (s1; ··· ; sr ) 2 (N ) , such that s1 ≥ 2 . Lemma: For all sequences (s1; ··· ; sr ), we have: s1;··· ;sr s1;··· ;sr−1 sr ∆−(He ) = He · J ; s −s where J (z) = z and ∆−(f )(z) = f (z − 1) − f (z). Heuristic: Bes1;··· ;sr (z) = Multiple (Divided) Bernoulli Polynomials = He−s1;··· ;−sr (z) : bes1;··· ;sr = Multiple (Divided) Bernoulli Numbers = He−s1;··· ;−sr (0) : Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function 2 The Form of a Multiple Bernoulli Polynomial 3 The General Reflexion Formula of Multiple Bernoulli Polynomial 4 An Example of Multiple Bernoulli Polynomial Main Goal We want to define Bes1;··· ;sr (z) such that: their properties are similar to Hurwitz Multiple Zeta Functions' properties. their properties generalize these of Bernoulli polynomials. Main Goal: Find some polynomials Bes1;··· ;sr such that: 8 s Bs+1(z) > Be (z) = , where s ≥ 0 ; > s + 1 > <> s1;··· ;sr s1;··· ;sr−1 sr ∆ (Be )(z) = Be (z) · z , where r ≥ 2 , and s1; ··· ; sr ≥ 0 ; > > > : the Bes1;··· ;sr are multiplied by the stuffle product. A particular solution of the main goal Lemma: The familly of difference equations ∆ (Bes1;··· ;sr )(z) = Bes1;··· ;sr−1 (z) · z sr s1;··· ;sr has a unique familly of solutions Be0 2 zC[z] , which are multiplied by the stuffle. Sketch of proof: ker ∆ \ zC[z] = f0g . Examples: p zX X p X e − 1 Beeg X (z) = Be (z) = : 0 0 p! eX − 1 p≥0 p q X +Y Y X p;q X Y Beeg (z) − Beeg (z) Beeg X ;Y (z) = Be (z) = 0 0 : 0 0 p! q! eX − 1 p;q≥0 The Form of a Multiple Bernoulli Polynomial s1;··· ;sr • Notations: The previous family Be0 is denoted by Be0 . In general, a family Ms1;··· ;sr is denoted by M• and called a mould . r s ;··· ;s X s ;··· ;s s ;··· ;s Mould product: M• = A• × B• () M 1 r = A 1 k B k+1 r . k=0 Theorem: • The solutions Be (z), valued in C[z] and multiplied by the stuffle product, to the difference equation ∆ (Bes1;··· ;sr )(z) = Bes1;··· ;sr−1 (z) · z sr satisfying B (z) Bes (z) = s+1 have an exponential generating function Beeg •(z) of the s + 1 form • • • Beeg (z) = beeg × Beeg0 (z) ; where: 8 • beeg is valued in C : > < The beeg •'s are multiplied by the stuffle product. 1 1 > beeg X = − : : eX − 1 X Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function 2 The Form of a Multiple Bernoulli Polynomial 3 The General Reflexion Formula of Multiple Bernoulli Polynomial 4 An Example of Multiple Bernoulli Polynomial • A New Expression of the Mould Beeg0 (z) New Goal: • • • • From Beeg (z) = beeg × Beeg0 (z), determine a suitable mould be , or beeg • such that the reflexion formula is a nice generalization of n (−1) Bn(1 − z) = Bn(z) ; n 2 N : Definition: z(X1+···+Xr ) X ;··· ;X e Sing 1 r (z) = ; for all z 2 C. (eX1 − 1)(eX1+X2 − 1) ··· (eX1+···+Xr − 1) Property: ×−1 • • • Beeg0 (z) = Sing (0) × Sing (z) : Key Point of the Reflexion Formula of Sing •(z) −• −!1;··· ;−!r Notations: M :(!1; ··· ;!r ) 7−! M . − • !r ;··· ;!1 M :(!1; ··· ;!r ) 7−! M . Lemma: Let Sg • be the mould defined by: Sg X1;··· ;Xr = (−1)r . For all z 2 C, we have: − ×−1 − ×−1 Sing −•(0) = Sing • (0) × Sg • , Sing −•(1 − z) = Sing • (z) . 1 1 Sketch of proof: Use + + 1 = 0 . eX − 1 e−X − 1 Examples: Sing −X ;−Y (0) = Sing X ;Y (0) + Sing X +Y (0) + Sing X (0) + 1 : Sing −X ;−Y (1 − z) = Sing X ;Y (z) + Sing X +Y (z) : Reflexion Formula for Multiple Bernoulli Polynomials Corollary: For all z 2 C, we have − −• • • • Beeg0 (1 − z) × Beeg0 (z) = 1 + I ; 1 if r = 0 : 1 if r = 1 : where 1X1;··· ;Xr = and I X1;··· ;Xr = 0 if r > 0 : 0 if r 6= 1 : Theorem: For all z 2 C, we have: − − • • Beeg −•(1 − z) × Beeg (z) = beeg −• × (1• + I •) × beeg : Outline 1 Reminders Reminders on Bernoulli Polynomials and Numbers Reminders on Riemann Zeta Function and Hurwitz Zeta Function 2 The Form of a Multiple Bernoulli Polynomial 3 The General Reflexion Formula of Multiple Bernoulli Polynomial 4 An Example of Multiple Bernoulli Polynomial A Mould Equation − −• Goal: Solve the mould equation X eeg • × (1• + I •) × X eeg = 1• . • • • Consequence: Define Beeg = X eeg × Beeg0 , so that: − • Beeg −•(1 − z) × Beeg (z) = 1• : Definition: p The mould Sg • is defined by: p (−1)r 2r Sg X1;··· ;Xr = 22r r Resolution of the Mould Equation Property: − −• Any solution X eeg • of X eeg • × (1• + I •) × X eeg = 1• comes from a mould Zeg • satisfying: − ( −• Zeg • + Zeg = 0• ; Zeg • is \primitive" , p and is given by: X eeg • = Exp(Zeg •) × Sg • : Sketch of proof: group-like = Exp(primitive) . Definition of Multiple Bernoulli Polynomials and Numbers The mould Zeg • defined by 8 ; < Zeg := 0 ; (−1)r−1 Zeg X1;··· ;Xr := Zeg X1+···+Xr ; : r 1 1 1 where Zeg X = − + , satifies the required conditions: eX − 1 X 2 − ( −• Zeg • + Zeg = 0 ; Zeg • is \primitive" , Definition: The moulds Beeg •(z) and beeg • are defined by: p Beeg •(z) = Exp(Zeg •) × Sg • × (Sing •(0))×−1 × Sing •(z) : p beeg • = Exp(Zeg •) × Sg • The coefficients Be•(z) and Be• of these exponential generating series are called respectively the Multiple Bernoulli Polynomials and Numbers. one out of four Multiple Bernoulli Numbers is null. one out of two diagonals is \constant". cross product around the zeros are equals : 28800 · 127008 = 604802 . \symmetrie" relatively to p = q . Table of Multiple Bernoulli Numbers in length 2 bep;q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of two diagonals is \constant". cross product around the zeros are equals : 28800 · 127008 = 604802 . \symmetrie" relatively to p = q . Table of Multiple Bernoulli Numbers in length 2 bep;q p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 3 1 1 1 q = 0 − 0 0 − 0 8 12 120 252 1 1 1 1 1 1 1 q = 1 − − − 24 288 240 2880 504 6048 480 1 1 1 q = 2 0 0 − 0 0 240 504 480 1 1 1 1 1 1 1 q = 3 − − − − 240 2880 504 28800 480 60480 264 1 1 1 q = 4 0 − 0 0 − 0 504 480 264 1 1 1 1 1 1 691 q = 5 − − − 504 6048 480 60480 264 127008 65520 one out of four Multiple Bernoulli Numbers is null.
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