Domains and a (new ?) family of entire functions. Symmetrizations - PowerPoint PPT Presentation

Domains and a (new ?) family of entire functions. Symmetrizations and Newton-Girard formula. V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ o et al. Collaboration at various stages of the work and in the framework of the Project Evolution Equations in Combinatorics and Physics : N. Behr, K. A. Penson, C. Tollu. CIP, 08 October 2019 1/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 1 / 24

Plan 3 Introduction 15 Continuing the ladder 4 Explicit construction of Li 16 On the right: freeness without 6 Li From Noncommutative Diff. monodromy Eq. 17 A useful property 8 Properties of the extended Li 18 Left and then right: the arrow Li (1) 9 Passing to harmonic sums • H w , w ∈ Y ∗ 19 Sketch of the proof (pictorial) 10 Global and local domains 20 Concluding remarks 11 Properties of the domains 2/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 2 / 24

Introduction The aim of this quick talk is to explain how to extend polylogarithms � z n 1 Li ( s 1 , . . . s r ) = (1) n s 1 1 . . . n s r r n 1 > n 2 >... n r > 0 They are a priori coded by lists ( s 1 , . . . s r ) but, when s i ∈ N + , they admit an iterated integral representation and are better coded by words with letters in X = { x 0 , x 1 } . We will use the one-to-one correspondences. + ↔ x s 1 − 1 x 1 . . . x s r − 1 ( s 1 , . . . , s r ) ∈ N r x 1 ∈ X ∗ x 1 ↔ y s 1 . . . y s r ∈ Y ∗ (2) 0 0 ere and, for ℜ ( s ) > 1, one has Li ( s )[1] = ζ ( s ) Li ( s )[ z ] is Jonqui` 0 ) = log n ( z ) Completed by Li ( x n this provides a family of independant n ! functions admitting an analytic continuation on the cleft plane � C \ (] − ∞ , 0] ∪ [1 , + ∞ [) or C \ { 0 , 1 } . 3/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 3 / 24

Explicit construction of Li Given a word w , we note | w | x 1 the number of occurrences of x 1 within w  1 Ω if w = 1 X ∗   � z  0 ( u ) ds 0 α s if w = x 1 u α z � z 1 − s 0 ( w ) = (3) 1 α s 0 ( u ) ds w = x 0 u and | u | x 1 = 0  if  � z  s 0 α s 0 ( u ) ds if w = x 0 u and | u | x 1 > 0 . s Of course, the third line of this recursion implies 0 ) = log ( z ) n α z 0 ( x n n ! one can check that (a) all the integrals (although improper for the fourth line) are well defined (b) the series S = � w ∈ X ∗ α z 0 ( w ) w satisfies (4). We then have α z 0 = Li . 4/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 4 / 24

x 3 x 0 x 2 x 2 x 2 x 1 x 2 x 3 x 1 x 0 x 1 x 0 x 1 x 0 0 x 1 1 x 0 1 1 0 0 x 2 x 0 x 1 x 1 x 0 x 2 1 0 x 1 x 0 1 X ∗ As an example, we compute some coefficients log ( z ) n ( − log (1 − z )) n � Li | x n � Li | x n 0 � = ; 1 � = n ! n ! z n � � Li | x 0 x 1 � = Li 2 ( z ) = ; � Li | x 1 x 0 � = � Li | x 1 ⊔ ⊔ x 0 − x 0 x 1 � ( z ) n 2 n ≥ 1 z n � Li | x 2 � 0 x 1 � = Li 3 ( z ) = ; � Li | x 1 x 0 � = ( − log (1 − z )) log ( z ) − Li 2 ( z ) n 3 n ≥ 1 z n 1 � Li | x r − 1 � Li | x 2 ⊔ x 0 x 1 ) + x 0 x 2 � x 1 � = Li r ( z ) = ; 1 x 0 � = � Li | ( x 1 ⊔ ⊔ x 0 ) − ( x 1 ⊔ 1 � ⊔ x 1 ⊔ 0 n r 2 n ≥ 1 5/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 5 / 24

Li From Noncommutative Diff. Eq. The generating series S = � w ∈ X ∗ Li ( w ) satisfies (and is unique to do so)  d ( S ) = ( x 0 x 1 z + 1 − z ) . S   (4)  z ∈ Ω S ( z ) e − x 0 log ( z ) = 1 H (Ω) �  lim z → 0 � X � � with X = { x 0 , x 1 } . This is, up to the sign of x 1 , the solution G 0 of Drinfel’d [2] for KZ3. We define this unique solution as Li . All Li w are C - and even C ( z )-linearly independant (see CAP 17 Linear independance without monodromy [5]). 6/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 6 / 24

Domain of Li (definition) In order to extend indexation of Li to series, we define Dom ( Li ; Ω) (or Dom ( Li )) if the context is clear) as the set of series S = � n ≥ 0 S n (decomposition by homogeneous components) such that � n ≥ 0 Li S n ( z ) converges unconditionally for compact convergence in Ω. One sets � Li S ( z ) := Li S n ( z ) (5) n ≥ 0 Starting the ladder Li • ( C � X � , ⊔ C { Li w } w ∈ X ∗ ⊔ , 1 X ∗ ) Li (1) ( C � X � , ⊔ ⊔ , 1 X ∗ )[ x ∗ 0 , ( − x 0 ) ∗ , x ∗ • 1 ] C Z { Li w } w ∈ X ∗ Examples 1 ( z ) = (1 − z ) − 1 , Li α x ∗ 1 ( z ) = z α (1 − z ) − β Li x ∗ 0 ( z ) = z , Li x ∗ 0 + β x ∗ 7/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 7 / 24

Properties of the extended Li Proposition With this definition, we have 1 Dom ( Li ) is a shuffle subalgebra of C � � X � � and so is Dom rat ( Li ) := Dom ( Li ) ∩ C rat � � X � � 2 For S , T ∈ Dom ( Li ), we have ⊔ T = Li S . Li T Li S ⊔ Examples and counterexamples For | t | < 1, one has ( tx 0 ) ∗ x 1 ∈ Dom ( Li , D ) ( D being the open unit slit disc and Dom ( Li , D ) defined similarly), whereas x ∗ 0 x 1 / ∈ Dom ( Li , D ). Indeed, we have to examine the convergence of � 0 x 1 ( z ), but, for n ≥ 0 Li x n 0 x 1 ( z ) ∈ R and therefore, for these values z ∈ ]0 , 1[, one has 0 < z < Li x n � 0 x 1 ( z ) = + ∞ . Contrariwise one can show that, for | t | < 1, n ≥ 0 Li x n Li ( tx 0 ) ∗ x 1 ( z ) = � z n n ≥ 1 n − t 8/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 8 / 24

Passing to harmonic sums H w , w ∈ Y ∗ Polylogarithms having a removable singularity at zero The following proposition helps us characterize their indices. Proposition Let f ( z ) = � L | P � = � w ∈ X ∗ � P | w � Li w . The following conditions are equivalent i) f can be analytically extended around zero ii) P ∈ C � X � x 1 ⊕ C . 1 X ∗ We recall the expansion (for w ∈ X ∗ x 1 ⊔ { 1 X ∗ } , | z | < 1) � Li w ( z ) H π Y ( w ) ( N ) z N 1 − z = (6) N ≥ 0 9/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 9 / 24

Global and local domains This proposition and the lemma lead us to the following definitions. Global domains .– 1 Let ∅ � = Ω ⊂ � B (with B = C { 0 , 1 } ), we define Dom Ω ( Li ) ⊂ C � � X � � to be the set of series S = � n ≥ 0 S n (with S n = � | w | = n � S | w � w each homogeneous component) such that � n ∈ N Li S n is unconditionally convergent for the compact convergence (UCC) [4]. As examples, we have Ω 1 , the doubly cleft plane then Dom ( Li ) := Dom Ω 1 ( Li ) or Ω 2 = � B Local domains around zero (fit with H -theory) .– 2 Here, we consider series S ∈ ( C � � x 1 ⊕ C 1 X ∗ ) (i.e. supp ( S ) ∩ Xx 0 = ∅ ). � X � We consider radii 0 < R ≤ 1, the corresponding open discs D R = { z ∈ C | | z | < R } and define � { S = Σ n ≥ 0 S n ∈ ( C � � x 1 ⊕ C 1 Ω ) | Dom R ( Li ) := � X � Li S n (UCC) in D R } n ∈ N Dom loc ( Li ) := ∪ 0 < R ≤ 1 Dom R ( Li ) . 10/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 10 / 24

Properties of the domains Theorem A 1 For all ∅ � = Ω ⊂ � B , Dom Ω ( Li ) is a shuffle subalgebra of C � � X � � and so are the Dom R ( Li ). 2 R �→ Dom R ( Li ) is strictly decreasing for R ∈ ]0 , 1]. 3 All Dom R ( Li ) and Dom loc ( Li ) are shuffle subalgebras of C � � X � � and π Y ( Dom loc ( Li )) is a stuffle subalgebra of C � � Y � � . 4 Let T ( z ) = � N ≥ 0 a N z N be a Taylor series i.e. such that lim sup N → + ∞ | a N | 1 / N = B < + ∞ , then the series � a N ( − ( − x 1 ) + ) ⊔ ⊔ N S = (7) N ≥ 0 is summable in C � � (with sum in C � � X � � x 1 � � ) and S ∈ Dom R ( Li ) with 1 R = B +1 and Li S = T ( z ). 11/24 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Domains and a (new ?) family of entire functions. o et al. Collaboration at various stages of the work and in the framework of the CIP, 08 October 2019 11 / 24