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Functional coefficients in solutions of non commutative differential equations

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43 pages
Functional coefficients in solutions of non-commutative differential equations Matthieu Deneufchatel, G. H. E. Duchamp, V. Hoang Ngoc Minh Laboratoire d'Informatique de Paris Nord, Universite Paris 13 Journees Nationales de Calcul Formel, 14-18 novembre 2011

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Functional coefficients in solutions of non-commutative
differential equations
Matthieu Deneufchaˆtel, G. H. E. Duchamp, V. Hoang Ngoc Minh
Laboratoire d’Informatique de Paris Nord,
Universit´e Paris 13
Journ´ees Nationales de Calcul Formel,
14-18 novembre 2011Outline
1 Motivation
2 Main Theorem
3 Examples
Polylogarithms
Counterexample
Hyperlogarithms
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 2 / 24Motivation
Outline
1 Motivation
2 Main Theorem
3 Examples
Polylogarithms
Counterexample
Hyperlogarithms
M. Deneufchˆatel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 3 / 24Motivation
Riemann ζ function :
X 1
ζ(s) = .
sn
n≥1
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 4 / 24Motivation
Riemann ζ function :
X 1
ζ(s) = .
sn
n≥1
Generalization (in view of multiplications) : Polyzetas
X 1
ζ(s) = .ss1 kn ...n1 kn >···>n >01 k
M. Deneufchˆatel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 4 / 24Motivation
Riemann ζ function :
X 1
ζ(s) = .
sn
n≥1
Generalization (in view of multiplications) : Polyzetas
X 1
ζ(s) = .ss1 kn ...n1 kn >···>n >01 k
Polyzetas are values of polylogs at 1 :
ζ(s) = Li (1).s
Polylogs can be manipulated as shuffles : algebra structure.
(Non commutative) Differential equation :

d 1 1
Drinfel’d equation T(z) = x + x T(z).0 1
dz z 1−z
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 4 / 24Motivation
Bailey & Borwein & Girgensohn
1994, Cartier 2002,
Zagier 1994, Racinet 2002,
Flajolet & Salvy 1998, Ecalle 2003,
Minh & Petitot & Van Der Cresson & Fischler & Rivoal
Hoeven 1998, 2006.
Waldshmidt 2000,
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 5 / 24Motivation
Bailey & Borwein & Girgensohn
1994, Cartier 2002,
Zagier 1994, Racinet 2002,
Flajolet & Salvy 1998, Ecalle 2003,
Minh & Petitot & Van Der Cresson & Fischler & Rivoal
Hoeven 1998, 2006.
Waldshmidt 2000,
Examples of relations between Multiple Zeta Values (M. Bigotte) :
6117808 6ζ(7,5) =− ζ(2) −7ζ(10,2)+28ζ(5)ζ(7)+14ζ(3)ζ(9)
2627625
93976 96ζ(9,3) =− ζ(2) − ζ(10,2)+12ζ(5)ζ(7)+9ζ(3)ζ(9)
79625 2
9 388112 6ζ(9,3) = ζ(7,5)+ ζ(2) −6ζ(5)ζ(7)
14 1226225
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 5 / 24Main Theorem
Outline
1 Motivation
2 Main Theorem
3 Examples
Polylogarithms
Counterexample
Hyperlogarithms
M. Deneufchˆatel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 6 / 24Main Theorem
Notations
If X is an alphabet and k a field,
∗X denotes the set of words with letters in X ;
khXi is the algebra of non commutative polynomials with coefficients
in k,
khhXii the algebra of non commutative series with coefficients in k ;
If S is a (non commutative) series and w a word, hS|wi denotes the
coefficient of S on w.
M. Deneufchaˆtel (LIPN - Universit´e Paris 13) Functional coefficients 14-18/11/2011 7 / 24