Mathematical renormalization in quantum electrodynamics via noncommutative generating series

TL;DR

This paper explores a noncommutative formal power series approach to understand the combinatorial structure of renormalization in quantum electrodynamics, especially at singularities of related differential equations.

Contribution

It introduces a novel application of noncommutative generating series to analyze the combinatorial aspects of QED renormalization at singular points.

Findings

Provides a new mathematical framework for renormalization analysis.

Identifies combinatorial structures underlying QED singularities.

Enhances understanding of differential equations in quantum field theory.

Abstract

In this work, we focus on on the approach by noncommutative formal power series to study the combinatorial aspects of the renormalization at the singularities in $\{0,1,+\infty\}$ of the solutions of nonlinear differential equations involved in quantum electrodynamics.

Equations

$$\displaystyle\frac{dG(z)}{dz}=\biggl{(}\displaystyle\frac{x_{0}}{z}+\displaystyle\frac{x_{1}}{1-z}\biggr{)}G(z),$$

$$\zeta (s_1,\dots ,s_r)=\sum _{n_1>\dots >n_r>0}\frac{1}{n_1^{s_1}\dots n_r^{s_r}},$$

$${\mathrm{Li}}_{s_1,\dots ,s_r}(z)=\sum _{n_1>\dots >n_r>0}\frac{z^{n_1}}{n_1^{s_1}\dots n_r^{s_r}}$$

$$\zeta (s_1,\dots ,s_r)=\underset{z\to 1}{\lim }{\mathrm{Li}}_{s_1,\dots ,s_r}(z)=\underset{N\to +\infty }{\lim }{\mathrm{H}}_{s_1,\dots ,s_r}(N).$$

$$(\{1{\}}^k,s_{k+1},\dots ,s_r)\leftrightarrow y_1^k{y}_{s_{k+1}}\dots y_{s_r}{\rightleftharpoons }_{{\pi }_Y}^{{\pi }_X}{x}_1^k{x}_0^{s_{k+1}-1}{x}_1\dots x_0^{s_r-1}{x}_1.$$

$$\mathrm{L}=\prod _{l\in \mathcal{L}ynX}\exp ({\mathrm{Li}}_{S_l}{P}_l)$$

$$Z_{{}_{{}^{\bigsqcup \bigsqcup }}}=\prod _{l\in \mathcal{L}ynX,l\ne x_0,x_1}\exp (\zeta (S_l)P_l)$$

$$\displaystyle\displaystyle\lim_{z\rightarrow 1}\exp\biggl{[}y_{1}\log\frac{1}{1-z}\biggr{]}\pi_{Y}\mathrm{L}(z)=Z_{1},$$

$$\displaystyle\displaystyle\prod_{l\in{\mathcal{L}yn}Y,l\neq y_{1}}^{\searrow}\exp(\zeta(\Sigma_{l})\Pi_{l})=\exp\biggl{[}\sum_{k\geq 2}-\zeta(k)\displaystyle\frac{(-y_{1})^{k}}{k}\biggr{]}\pi_{Y}\displaystyle\prod_{l\in{\mathcal{L}yn}X,l\neq x_{0},x_{1}}^{\searrow}\exp(\zeta(S_{l})P_{l}).$$

$$z(1-z)\ddot{y}(z)+[t_2-(t_0+t_1+1)z]\dot{y}(z)-t_0{t}_{1}y(z)=0.$$

$$\left(\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\end{array}\right)=\left(\frac{M_0}{z}+\frac{M_1}{1-z}\right)\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right),$$

$$M_0=-\left(\begin{array}{cc}0& 0\\ t_0{t}_1& t_2\end{array}\right)$$

$$\dot{q}(z)=A_0(q)\frac{1}{z}+A_1(q)\frac{1}{1-z}$$

$$A_0=-(t_0{t}_1{q}_1+t_2{q}_2)\frac{\partial }{\partial q_2}$$

$$\frac{\partial }{\partial q_1}(q)=\frac{\partial }{\partial q_1}\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)$$

$$\dot{y}(z)+k_{1}y(z)+k_2{y}^2(z)=u_1(z).$$

$$\dot{q}(z)=A_0(q)+A_1(q)u_1(z)$$

$$A_0=-(k_{1}q+k_2{q}^2)\frac{\partial }{\partial q}$$

$$\ddot{y}(z)+a\dot{y}(z)+by(z)+c{y}^3(z)=u_1(z).$$

$$\dot{q}(z)=A_0(q)+A_1(q)u_1(z),$$

$$A_0=-(a{q}_2+b^2{q}_1+c{q}_1^3)\frac{\partial }{\partial q_2}+q_2\frac{\partial }{\partial q_1}$$

$${\partial }_z^{2}x(z)-\gamma [1+x(z{)}^2]{\partial }_{z}x(z)+x(z)=g\cos (\omega z)$$

$${\partial }_{z}x(z)=\gamma [1+x(z{)}^2/3]x(z)-{\int }_{z_0}^{z}x(s)ds+\frac{g}{\omega }\sin (\omega z)+C.$$

$${\partial }_z^{2}y(z)=\gamma [{\partial }_{z}y(z)+({\partial }_{z}y(z){)}^3/3]-y(z)+u_1(z)$$

$${\partial }_{z}q(z)=[A_0{u}_0(z)+A_1{u}_1(z)](q)$$

$$A_1=\frac{\partial }{\partial q_2}$$

$$\forall w\in X^{\ast },$$

$$\forall x,y\in X,\forall u,v\in X^{\ast },$$

$${\Delta }_{\mathtt{conc}}(w)=\sum _{u,v\in X^{\ast },uv=w}u\otimes v$$

$$⟨{\Delta }_{\mathtt{conc}}(w)\mid u\otimes v⟩=⟨w\mid uv⟩$$

$${\mathcal{H}}_{{}_{{}^{\bigsqcup \bigsqcup }}}=(\mathbb{Q}⟨X⟩,\mathtt{conc},1_{X^{\ast }},{\Delta }_{{}_{{}^{\bigsqcup \bigsqcup }}},ϵ),$$

Full text

1 Introduction

During the last century, the functional expansions were common in physics as well as in engineering and have been developped, by Tomonaga, Schwinger and Feynman dyson , to represent the dynamical systems in quantum electrodynanics. The main difficulty of this approach is the divergence of these expansions at the singularity [math] or at $+\infty$ (see BBM ) and leads to the problems of regularization and renormalization which can be solved by combinatorial technics : Feynman diagrams FH and their siblings interface ; lemurakami , noncommutative formal power series fliess1 , trees CK .

Recently, in the same vein, and based, on the one hand, on the shuffle and quasi-shuffle bialgebras BDHHT , the combinatorics of noncommutative formal power series was intensively amplified for the asymptotic analysis of dynamical systems with three regular singularities in222Any differential equation with singularities in $\{a,b,c\}$ can be changed into a differential equation with singularities in $\{0,1,+\infty\}$ via an homographic transformation. $\{0,1,+\infty\}$ ; and, on the other hand with the monodromy and the Galois differential group of the Knizhnik-Zamolodchikov equation $KZ_{3}$ orlando ; acta i.e., the following noncommutative evolution equation333$x_{0}:=\phantom{-}{t_{1,2}}/{2\mathrm{i}\pi}$ and $x_{1}:=-{t_{2,3}}/{2\mathrm{i}\pi}$ are noncommuative variables and $t_{1,2},t_{2,3}$ belong to $\mathcal{T}_{3}=\{t_{1,2},t_{1,3},t_{2,3}\}$ satisfying the infinitesimal $3$-braid relations, i.e. $[t_{1,3},t_{1,2}+t_{2,3}]=[t_{2,3},t_{1,2}+t_{1,3}]=0$.

[TABLE]

the monoidal factorization facilitates mainly the renormalization and the computation of the associators444 They were introduced in QFT by Drinfel’d and it plays an important role for the still open problem of the effective determination of the polynomial invariants of knots and links via Kontsevich’s integral (see cartier ; lemurakami ) and $\Phi_{KZ}$, was obtained firstly, in lemurakami , with explicit coefficients which are polyzetas and regularized polyzetas (see acta ; VJM for the computation of the other involving only convergent polyzetas as local coordinates, and for algorithms regularizing divergent polyzetas). via the universal one, i.e. $\Phi_{KZ}$ of Drinfel’d acta .

In fact, these associators are noncommutative formal power series on two variables and regularize the Chen generating series of the differential forms admitting singularities at [math] or at $1$ along the integration paths on the universal covering of ${\mathbb{C}}$ without points [math] and $1$ (i.e. $\widetilde{{\mathbb{C}}\setminus\{0,1\}}$). Their coefficients are, up to a multiple of powers of $2\mbox{i}\pi$, polynomial on polyzetas, i.e. the following real numbers555$s_{1}+\ldots+s_{r}$ is the weight of $\zeta(s_{1},\ldots,s_{r})$ , i.e. the weight of the composition $(s_{1},\ldots,s_{r})$. lemurakami ; hoffman0 ; zagier

[TABLE]

and these numbers admit a natural structure of algebra over the rational numbers deduced from the combinatorial aspects of the shuffle and quasi-shuffle Hopf algebras. It is conjectured that this algebra is ${\mathbb{N}}$-graded666One of us proposed a proof in acta ; VJM .. More precisely, for $s_{1}\geq 2,s_{2},\ldots,s_{r}\geq 1$, the polyzeta $\zeta(s_{1},\ldots,s_{r})$ can be obtained as the limit of the polylogarithm $\operatorname{Li}_{s_{1},\ldots,s_{r}}(z)$, for $z\rightarrow 1$, and of the harmonic sum $\mathrm{H}_{s_{1},\ldots,s_{r}}(N)$, for $N\rightarrow+\infty$ :

[TABLE]

Then, by a theorem of Abel, one has

[TABLE]

Since the algebras of polylogarithms and of harmonic sums are isomorphic to the shuffle algebra $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ and quasi-shuffle algebra ({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}}) respectively both admitting the Lyndon words ${\mathcal{L}yn}X$ over $X=\{x_{0},x_{1}\}$ and ${\mathcal{L}yn}Y$ over $Y=\{y_{i}\}_{i\geq 1}$, as transcendence bases (recalled in Section 2.1) then, by using

• •

The (one-to-one) correspondence between the combinatorial compositions, the words777Here, $X^{*}$ (resp. $Y^{*}$) is the monoid generated by $X$ (resp. $Y$) and its neutral element of is denoted by $1_{X^{*}}$ (resp. $1_{Y^{*}}$). in $Y^{*}$ and the words in $X^{*}x_{1}+1_{X^{*}}$, i.e.888Here, $\pi_{Y}$ is the adjoint of $\pi_{X}$ for the canonical scalar products where $\pi_{X}$ is the morphism of AAU $k\langle Y\rangle\rightarrow k\langle X\rangle$ defined by $\pi_{X}(y_{k})=x_{0}^{k-1}x_{1}$.

[TABLE]

• •

The ordering $x_{1}\succ x_{0}$ and $y_{1}\succ y_{2}\succ\ldots$ over $X$ and $Y$ respectively.

• •

The transcendence base $\{S_{l}\}_{l\in{\mathcal{L}yn}X}$ (resp. $\{\Sigma_{l}\}_{l\in{\mathcal{L}yn}Y}$) of $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ (resp. ({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}})) in duality999in a more precise way the $S$ and $\Sigma$ are the “Lyndon part” of the dual bases of the PBW expansions of the $P$ and the $\Pi$ respectively. with $\{P_{l}\}_{l\in{\mathcal{L}yn}X}$ (resp. $\{\Pi_{l}\}_{l\in{\mathcal{L}yn}Y}$), a base of the Lie algebra of primitive elements of the bialgebra101010$\epsilon$ is the “constant term” character. ${\mathcal{H}}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}=({\mathbb{Q}}\langle X\rangle,{\tt conc},1_{X^{*}},\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}},\epsilon)$ (resp. {\mathcal{H}}_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}=({\mathbb{Q}}\langle Y\rangle,{\tt conc},1_{Y^{*}},\Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;},\epsilon)) to factorize the following noncommutative generating series of polylogarithms, hormanic sums and polyzetas

[TABLE]

we then obtain two formal power series over $Y$, $Z_{1}$ and $Z_{2}$, such that

[TABLE]

Moreover, $Z_{1},Z_{2}$ are equal and stand for the noncommutative generating series of $\{\zeta(w)\}_{w\in Y^{*}-y_{1}Y^{*}}$, or $\{\zeta(w)\}_{w\in x_{0}X^{*}x_{1}}$, as one has $Z_{1}=Z_{2}=\pi_{Y}Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ cade ; acta ; VJM . This allows, by extracting the coefficients of the noncommutative generating series, to explicit the counter-terms eliminating the divergence of $\{\operatorname{Li}_{w}\}_{w\in x_{1}X^{*}}$ and of $\{\mathrm{H}_{w}\}_{w\in y_{1}Y^{*}}$ and this leads naturally to an equation connecting algebraic structures

[TABLE]

Identity (2) allows to compute the Euler-MacLaurin constants and the Hadamard finite parts associated to divergent polyzetas $\{\zeta(w)\}_{w\in y_{1}Y^{*}}$ and, by identifying local coordinates, to describe the graded core of $\ker\zeta$ by its algebraic generators.

In this paper, we will focus on the approach by noncommutative formal power series, adapted from fliess0 ; fliess1 , and explain how the results of cade ; acta ; VJM , allow to study the combinatorial aspects of the renormalization at the singularities in $\{0,1,+\infty\}$ of the solutions of linear differential equations (see Example 1 below) as well as the solutions of nonlinear differential equations (see Examples 2 and 3 below) described in Section 3.2 and involved in quantum electrodynamics.

Example 1 (Hypergeometric equation)

Let $t_{0},t_{1},t_{2}$ be parameters and

[TABLE]

Let $q_{1}(z)=-y(z)$ and $q_{2}(z)=(1-z)\dot{y}(z)$. One has

[TABLE]

where $M_{0}$ and $M_{1}$ are the following matrices

[TABLE]

Or equivalently,

[TABLE]

where $A_{0}$ and $A_{1}$ are the following parametrized linear vector fields

[TABLE]

acting by

[TABLE]

Example 2 (Harmonic oscillator)

Let $k_{1},k_{2}$ be parameters and

[TABLE]

which can be represented, with the same formalism as above, by the following state equations

[TABLE]

where $A_{0}$ and $A_{1}$ are the following vector fields

[TABLE]

Example 3 (Duffing’s equation)

Let $a,b,c$ be parameters and

[TABLE]

which can be represented by the following state equations

[TABLE]

where $A_{0}$ and $A_{1}$ are the following vector fields

[TABLE]

Example 4 (Van der Pol oscillator)

Let $\gamma,g$ be parameters and

[TABLE]

which can be tranformed into (with $C$ is some constant of integration)

[TABLE]

Setting $y=\int_{z_{0}}^{z}x(s)ds$ and $u_{1}(z)=g\sin(\omega z)/{\omega}+C$, it leads then to

[TABLE]

which can be represented by the following state equations (with $n=2$)

[TABLE]

where $A_{0}$ and $A_{1}$ are the following vector fields

[TABLE]

This approach by noncommutative formal power series is adequate for studying the algebraic combinatorial aspects of the asymptotic analysis at the singularities in $\{0,1,+\infty\}$, of the nonlinear dynamical systems described in Section 3.2 because

• •

The polylogarithms form a basis of an infinite dimensional universal Picard-Vessiot extension by means of these differential equations orlando ; DDMS and their algebra, isomorphic to the shuffle algebra, admits $\{\operatorname{Li}_{l}\}_{l\in{\mathcal{L}yn}X}$ as a transcendence basis,

• •

The harmonic sums generate the coefficients of the ordinary Taylor expansions of their solutions (when these expansions exist) cade and their algebra is isomorphic to the quasi-shuffle algebra admitting $\{\mathrm{H}_{l}\}_{l\in{\mathcal{L}yn}Y}$ as a transcendence basis,

• •

The polyzetas do appear as the fondamental arithmetical constants involved in the computations of the monodromies FPSAC98 ; SLC43 , the Kummer type functional equations FPSAC99 ; SLC43 , the asymptotic expansions of solutions cade ; acta and their algebra is freely generated by the polyzetas encoded by irreducible Lyndon words acta .

Hence, a lot of algorithms can be deduced from these facts and more general studies will be proceeded in BDHHT ; DDMS . The organisation of this paper is the following

• •

In Section 2, we will give algebraic and analytic foundations, i.e. the combinatorial Hopf algebra of shuffles and the indiscernability respectively, for polyzetas.

• •

These will be exploited, in Section 3, to expand solutions, of nonlinear dynamical systems with singular inputs and their ordinary and functional differentiations.

2 Fundation of the present framework

2.1 Background about combinatorics of shuffle and stuffle bialgebras

Schützenberger’s monoidal factorization

Let ${\mathbb{Q}}\langle X\rangle$ be equipped by the concatenation and the shuffle defined by

[TABLE]

or by their dual co-products, $\Delta_{\tt conc}$ and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, defined by, for $w\in X^{*}$ and $x\in X$,

[TABLE]

$\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ is then extended to a conc-morphism ${\mathbb{Q}}\langle X\rangle\rightarrow{\mathbb{Q}}\langle X\rangle\otimes{\mathbb{Q}}\langle X\rangle$. These two comultiplications satisfy, for any $u,v,w\in X^{*}$,

[TABLE]

One gets two mutually dual bialgebras

[TABLE]

After a theorem by Radford radford , ${\mathcal{L}yn}X$ forms a transcendence basis of $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ and it can be completed then to the linear basis $\{w\}_{w\in X^{*}}$ which is auto-dual :

[TABLE]

But the elements $l\in{\mathcal{L}yn}X-X$ are not primitive, for $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, and then ${\mathcal{L}yn}X$ does not constitute a basis for ${\cal L}ie_{{\mathbb{Q}}}\langle X\rangle$. Chen, Fox and Lyndon lyndon constructed $\{P_{w}\}_{w\in X^{*}}$, so-called the Poincaré-Birkhoff-Witt-Lyndon basis, for ${\cal U}({\cal L}ie_{{\mathbb{Q}}}\langle X\rangle)$ as follows

[TABLE]

where here $\succ$ stands for the lexicographic (strict) ordering defined111111In here, the order relation $\succ$ on $X^{*}$ is defined by, for any $u,v\in X^{*}$, $u\succ v$ iff $u=vw$ with $w\in X^{+}$ else there are $w,w_{1},w_{2}\in X^{*}$ and $a\succ b\in X$ such that $u=waw_{1}$ and $v=wbw_{2}$. by $x_{0}\prec x_{1}$. Schützenberger constructed bases for $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ defined by duality as follows :

[TABLE]

and obtained the transcendence and linear bases, $\{S_{l}\}_{l\in\in{\mathcal{L}yn}X},\{S_{w}\}_{w\in X^{*}}$, as follows

[TABLE]

After that, Mélançon and Reutenauer reutenauer proved that, for any $w\in X^{*}$,

[TABLE]

where $|w|_{X}=(|w|_{x})_{x\in X}$ is the family of all partial degrees (number of times a letter occurs in a word). In other words, the elements of the bases $\{S_{w}\}_{w\in X^{*}}$ and $\{P_{w}\}_{w\in X^{*}}$ are lower and upper triangular respectively and they are of multihomogeneous (all the monomials have the same partial degrees).

Example 5 (of $\{P_{w}\}_{w\in X^{*}}$ and $\{S_{w}\}_{w\in X^{*}}$, hoangjacoboussous )

Let $X=\{x_{0},x_{1}\}$ with $x_{0}\prec x_{1}$.

[TABLE]

Then, Schützenberger’s factorization of the diagonal series ${\cal D}_{X}$ follows reutenauer

[TABLE]

Extended Schützenberger’s monoidal factorization

Let us define the commutative product over ${\mathbb{Q}}\langle Y\rangle$, denoted by $\mu$, as follows

[TABLE]

or by its associated coproduct, $\Delta_{\mu}$, defined by

[TABLE]

satisfying,

[TABLE]

Let ${\mathbb{Q}}\langle Y\rangle$ be equipped by

The concatenation (or by its associated coproduct, $\Delta_{\tt conc}$). 2. 2.

The shuffle product, i.e. the commutative product defined by fliess1

[TABLE]

or with its associated coproduct, $\Delta_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$, defined, on the letters, by

[TABLE]

and extended by morphism. It satisfies

[TABLE] 3. 3.

The quasi-shuffle product, i.e. the commutative product defined by hoffman

[TABLE]

or with its associated coproduct, \Delta_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}, defined, on the letters, by

[TABLE]

and extended by morphism. It satisfies

[TABLE]

Hence, with the counit $\tt e$ defined by, for any $P\in{\mathbb{Q}}\langle Y\rangle$, ${\tt e}(P)=\langle P\mid 1_{Y^{*}}\rangle$, one gets two pairs of mutually dual bialgebras

[TABLE]

By the CQMM theorem (see BDHHT ), the connected ${\mathbb{N}}$-graded, co-commutative Hopf algebra ${\mathcal{H}}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ is isomorphic to the enveloping algebra of the Lie algebra of its primitive elements which is equal to ${{\mathcal{L}}ie}_{\mathbb{Q}}\langle Y\rangle$ :

[TABLE]

Hence, let us consider lyndon

The PBW-Lyndon basis $\{p_{w}\}_{w\in Y^{*}}$ for ${\cal U}({{\mathcal{L}}ie}_{\mathbb{Q}}\langle Y\rangle)$ constructed recursively

[TABLE] 2. 2.

And, by duality121212The dual family of a basis lies in the algebraic dual which is here the space of noncommutative series, but as the enveloping algebra under consideration is graded in finite dimensions (here by the multidegree), these series are in fact (multihomogeneous) polynomials., the linear basis $\{s_{w}\}_{w\in Y^{*}}$ for $({\mathbb{Q}}\langle Y\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{Y^{*}})$, i.e.

[TABLE]

This basis can be computed recursively as follows reutenauer

[TABLE]

As in (8), one also has Schützenberger’s factorization for the diagonal series ${\cal D}_{Y}$

[TABLE]

Similarly, by the CQMM theorem, the connected ${\mathbb{N}}$-graded, co-commutative Hopf algebra {\mathcal{H}}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} is isomorphic to the enveloping algebra of

[TABLE]

where, for any $w\in Y^{*},\pi_{1}(w)$ is obtained as follows BDHHT ; acta

[TABLE]

Note that Equation (9) is equivalent to the following identity BDHHT ; acta ; VJM

[TABLE]

In particular, for any $y_{k}\in Y$, we have successively BDHHT ; acta ; VJM

[TABLE]

Hence, by introducing the new alphabet $\bar{Y}=\{\bar{y}\}_{y\in Y}=\{\pi_{1}(y)\}_{y\in Y}$, one has

[TABLE]

as one can prove through (12) that the endomorphism $y\mapsto\bar{y}$ is, in fact, an isomorphism

[TABLE]

By considering

The PBW-Lyndon basis $\{\Pi_{w}\}_{w\in Y^{*}}$ for {\cal U}(\mathrm{Prim}({\mathcal{H}}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}})) constructed recursively as follows acta

[TABLE] 2. 2.

And, by duality, the linear basis $\{\Sigma_{w}\}_{w\in Y^{*}}$ for ({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}}), i.e.

[TABLE]

This basis can be computed recursively as follows BDM ; acta

[TABLE]

In $(!)$, the sum is taken over all subsequences $\{k_{1},\ldots,k_{i}\}\allowbreak\subset\allowbreak\{1,\ldots,k\}$ and all Lyndon words $l_{1}\succeq\cdots\succeq l_{n}$ such that $(y_{s_{1}},\ldots,y_{s_{k}})\stackrel{{\scriptstyle*}}{{\Leftarrow}}(y_{s_{k_{1}}},\ldots,y_{s_{k_{i}}},\allowbreak l_{1},\ldots,l_{n})$, where $\stackrel{{\scriptstyle*}}{{\Leftarrow}}$ denotes the transitive closure of the relation on standard sequences, denoted by $\Leftarrow$ (see BDM ).

We also proved that, for any $w\in Y^{*}$, BDHHT ; acta ; VJM

[TABLE]

In other words, the elements of the bases $\{\Sigma_{w}\}_{w\in Y^{*}}$ and $\{\Pi_{w}\}_{w\in Y^{*}}$ are lower and upper triangular respectively and they are of homogeneous in weight.

We also get the extended Schützenberger’s factorization of ${\cal D}_{Y}$ BDHHT ; acta ; VJM

[TABLE]

Example 6 (of $\{\Pi_{w}\}_{w\in Y^{*}}$ and $\{\Sigma_{w}\}_{w\in Y^{*}}$, BDM )

[TABLE]

2.2 Indiscernability over a class of formal power series

Residual calculus and representative series

Definition 1

Let $S\in{\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$ (resp. ${\mathbb{Q}}\langle X\rangle$) and let $P\in{\mathbb{Q}}\langle X\rangle$ (resp. ${\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$). The left and right residual of $S$ by $P$ are respectively the formal power series $P\triangleleft S$ and $S\triangleright P$ in ${\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$ defined by $\langle P\triangleleft S\mid w\rangle=\langle S\mid wP\rangle$ (resp. $\langle S\triangleright P\mid w\rangle=\langle S\mid Pw\rangle$).

For any $S\in{\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$ (resp. ${\mathbb{Q}}\langle X\rangle$) and $P,Q\in{\mathbb{Q}}\langle X\rangle$ (resp. ${\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$), we straightforwardly get $P\triangleleft(Q\triangleleft S)=PQ\triangleleft S,(S\triangleright P)\triangleright Q=S\triangleright PQ$ and $(P\triangleleft S)\triangleright Q=P\triangleleft(S\triangleright Q)$.

In case $x,y\in X$ and $w\in X^{*}$, we get131313For any words $u,v\in X^{*}$, if $u=v$ then $\delta_{u}^{v}=1$ else [math]. $x\triangleleft(wy)=\delta_{x}^{y}w$ and $xw\triangleright y=\delta_{x}^{y}w$.

Lemma 1

(Reconstruction lemma)* Let $S\in{\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$. Then*

[TABLE]

Theorem 2.1

Le $\delta\in\mathfrak{Der}({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$. Moreover, we suppose that $\delta$ is locally nilpotent141414 $\phi\in End(V)$ is said to be locally nilpotent iff, for any $v\in V$, there exists $N\in{\mathbb{N}}$ s.t. $\phi^{N}(v)=0$.. Then the family $(t\delta)^{n}/{n!}$ is summable and its sum, denoted $\exp(t\delta)$, is is a one-parameter group of automorphisms of $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$.

Theorem 2.2

Let $L$ be a Lie series, i.e. $\Delta_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}(L)=L\hat{\otimes}1+1\hat{\otimes}L$. Let $\delta^{r}_{L},\delta^{l}_{L}$ be defined respectively by $\delta^{r}_{L}(P):=P\triangleleft L,\delta^{l}_{L}(P):=L\triangleright P$. Then $\delta^{r}_{L},\delta^{l}_{L}$ are locally nilpotent derivations of $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$. Hence, $\exp(t\delta^{r}_{L}),\exp(t\delta^{l}_{L})$ are one-parameter groups of $Aut({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ and $\exp(t{\delta^{r}_{L}})P=P\triangleleft\exp(tL),\exp(t{\delta^{l}_{L}})P=\exp(tL)\triangleright P$.

Example 7

Since $x_{1}\triangleleft$ and $\triangleright x_{0}$ are derivations and the polynomials $\{\Sigma_{l}\}_{l\in\in{\mathcal{L}yn}X-X}$ belong to $x_{0}\mathbb{Q}\langle X\rangle x_{1}$ then $x_{1}\triangleleft l=l\triangleright x_{0}=0$ and $x_{1}\triangleleft\check{S}_{l}=\check{S}_{l}\triangleright x_{0}=0$.

Theorem 2.3

Let $S\in{\mathbb{Q}}\langle\!\langle X\rangle\!\rangle$. The following properties are equivalent:

The left ${\mathbb{C}}$-module $Res_{g}(S)=\text{span}\{w\triangleleft S\mid w\in X^{*}\}$ is finite dimensional. 2. 2.

The right ${\mathbb{C}}$-module $Res_{d}(S)=\text{span}\{S\triangleright w\mid w\in X^{*}\}$ is finite dimensional. 3. 3.

There are matrices $\lambda\in\mathcal{M}_{1,n}(\mathbb{Q})$, $\eta\in\mathcal{M}_{n,1}(\mathbb{Q})$ and $\mu:X^{*}\longrightarrow\mathcal{M}_{n,n}$, such that

[TABLE]

A series that satisfies the items of Theorem 2.3 will be called representative series. This concept can be found in abe ; Duchamp ; hochschild ; DT2 . The two first items are in fliess0 ; hespel . The third can be deduced from ChariPressley ; Duchamp for example and it was used to factorize, for the first time, by Lyndon words, the output of bilinear and analytical dynamical systems respectively in IMACS0 ; hoangjacoboussous and to study polylogarithms, hypergeometric functions and associated functions in FPSAC95 ; FPSAC96 ; orlando . The dimension of $Res_{g}(S)$ is equal to that of $Res_{d}(S)$, and to the minimal dimension of a representation satisfying the third point of Theorem 2.3. This rank is then equal to the rank of the Hankel matrix of $S$, i.e. the infinite matrix $(\langle S\mid uv\rangle)_{u,v\in X}$ indexed by $X^{*}\times X^{*}$ so called Hankel rank151515 i.e. the dimension of $\mathrm{span}\{S\triangleright\Pi\mid\Pi\in{\mathbb{C}}\langle X\rangle\}$ (resp. $\mathrm{span}\{\Pi\triangleleft S\mid\Pi\in{\mathbb{C}}\langle X\rangle\}$). of $S$ fliess0 ; hespel . The triplet $(\lambda,\mu,\eta)$ is called a linear representation of $S$161616 The minimal representation of $S$ as being a representation of $S$ of minimal dimension. It can be shown that all minimal representations are isomorphic (see berstel ). . $S$ is called rational if it belongs to the closure by +, conc and star operation of proper elements171717 For any proper series $S$, i.e. $\langle S\mid 1_{X^{*}}\rangle=0$, the series $S^{*}=1+S+S^{2}+\ldots$ is called “star of $S$”. Any noncommutative power series is representative if and only if it is rational berstel ; schutz . These rationality properties can be expressed in terms of differential operators in noncommutative geometry Duchamp .

Background on continuity and indiscernability

Definition 2

(these ; cade ) Let ${\mathcal{H}}$ be a class of ${\mathbb{C}}\langle\!\langle X\rangle\!\rangle$ and $S\in{\mathbb{C}}\langle\!\langle X\rangle\!\rangle$.

$S$ is said to be continuous over ${\mathcal{H}}$ if for any $\Phi\in{\mathcal{H}}$, the following sum, denoted by $\langle S\mid\!\mid\Phi\rangle$, is absolutely convergent $\sum_{w\in X^{*}}\langle S\mid w\rangle\langle\Phi\mid w\rangle$.

The set of continuous power series over ${\mathcal{H}}$ will be denoted by ${\mathbb{C}}^{\mathrm{cont}}\langle\!\langle X\rangle\!\rangle$. 2. 2.

$S$ is said to be indiscernable over ${\mathcal{H}}$ if and only if, for any $\Phi\in{\mathcal{H}}$, $\langle S\mid\!\mid\Phi\rangle=0$.

Proposition 1

Let $S\in{\mathbb{C}}^{\mathrm{cont}}\langle\!\langle X\rangle\!\rangle$. ${\mathcal{H}}$ is a monoid containing $X$ and $\{e^{tx}\}_{x\in X}^{t\in{\mathbb{C}}}$.

If $S$ is indiscernable over ${\mathcal{H}}$ then for any $x\in X$, $x\triangleleft S$ and $S\triangleright x$ belong to ${\mathbb{C}}^{\mathrm{cont}}\langle\!\langle X\rangle\!\rangle$ and they are indiscernable over ${\mathcal{H}}$. 2. 2.

$S$* is indiscernable over ${\mathcal{H}}$ if and only if $S=0$.*

Proof

Of course, $x\triangleleft S$ and $S\triangleright x$ belong to ${\mathbb{C}}^{\mathrm{cont}}\langle\!\langle X\rangle\!\rangle$. Let us calculate $\langle x\triangleleft S\mid\!\mid\Phi\rangle=\langle S\mid\!\mid\Phi x\rangle$ and $\langle S\triangleright x\mid\!\mid\Phi\rangle=\langle S\mid\!\mid x\Phi\rangle$. Since $S$ is indiscernable over ${\mathcal{H}}$ and note that $x\Phi,\Phi x\in{\mathcal{H}}$ for evvery $x\in X;\Phi\in{\mathcal{H}}$, then

[TABLE]

Hence $x\triangleleft S$ and $S\triangleright x$ belong to ${\mathbb{C}}^{\mathrm{cont}}\langle\!\langle X\rangle\!\rangle$ are indiscernable over ${\mathcal{H}}$. 2. 2.

$S=0$ is indiscernable over ${\mathcal{H}}$. Conversely, if $S$ is indiscernable over ${\mathcal{H}}$ then by the previous point and by induction on the length of $w\in X^{*}$, $w\triangleleft S$ is indiscernable over ${\mathcal{H}}$. In particular, $\langle w\triangleleft S\mid\mathrm{Id}_{{\mathcal{H}}}\rangle=\langle S\mid w\rangle=0$. In other words, $S=0$.

2.3 Polylogarithms and harmonic sums

Structure of polylogarithms and of harmonic sums

Let $\Omega:={\mathbb{C}}-(]-\infty,0]\cup[1,+\infty[)$ and let ${\mathcal{C}}:={\mathbb{C}}[z,1/z,1/{1-z}]$. Note that the neutral element of ${\mathcal{C}}$, for the pointwise product, is $1_{\Omega}:\Omega\longrightarrow{\mathbb{C}}$ such that $z\longmapsto 1$.

One can check that $\operatorname{Li}_{s_{1},\ldots,s_{r}}$ is obtained as the iterated integral over the differential forms $\omega_{0}(z)={dz}/z$ and $\omega_{1}(z)={dz}/(1-z)$ and along the path $0\rightsquigarrow z$ IMACS  :

[TABLE]

By (1), $\operatorname{Li}_{s_{1},\ldots,s_{r}}$ is then denoted also by $\operatorname{Li}_{x_{0}^{s_{1}-1}x_{1}\ldots x_{0}^{s_{r}-1}x_{1}}$ or $\operatorname{Li}_{y_{s_{1}}\ldots y_{s_{r}-1}}$ FPSAC95 ; FPSAC96 ; SLC43 .

Example 8 (of $\operatorname{Li}_{2}=\operatorname{Li}_{x_{0}x_{1}}$)

[TABLE]

The definition of polylogarithms is extended over the words $w\in X^{*}$ by putting $\operatorname{Li}_{x_{0}}(z):=\log(z)$. The $\{\operatorname{Li}_{w}\}_{w\in X^{*}}$ are ${\mathcal{C}}$-linearly independent DDMS ; FPSAC98 ; SLC43 and then the following function, for $v=y_{s_{1}}\ldots y_{s_{r}}\in Y^{*}$, are also ${\mathbb{C}}$-linearly independent DDMS ; words03

[TABLE]

Proposition 2

(words03 )* By linearity, the following maps are isomorphisms of algebras*

[TABLE]

Theorem 2.4

(cade )* The Hadamard ${\mathcal{C}}$-algebra of $\{\mathrm{P}_{w}\}_{w\in Y^{*}}$ can be identified with that of $\{\mathrm{P}_{l}\}_{l\in{\mathcal{L}yn}Y}$. In the same way, the algebra of harmonic sums $\{\mathrm{H}_{w}\}_{w\in Y^{*}}$ with polynomial coefficients can be identified with that of $\{\mathrm{H}_{l}\}_{l\in{\mathcal{L}yn}Y}$.*

Let $\mathrm{L},\mathrm{P}$ and $\mathrm{H}$ be the noncommutative generating series of respectively $\{\operatorname{Li}_{w}\}_{w\in X^{*}},\allowbreak\{\mathrm{P}_{w}\}_{w\in X^{*}}$ and $\{\mathrm{H}_{w}(N)\}_{w\in Y^{*}}$, for $\mid\!z\!\mid<1$ and $N>1$ FPSAC98 ; words03  :

[TABLE]

Definition 3 (Polylogarithms and harmonic sums at negative multi-indices)

For any $s_{1},\ldots,s_{r}\in({\mathbb{N}})^{r}$, let us define GHN , for $\mid\!z\!\mid<1$ and $N>0$,

[TABLE]

The ordinary generating series, $\mathrm{P}_{-s_{1},\ldots,-s_{r}}(z)$, of $\{\mathrm{H}_{-s_{1},\ldots,-s_{r}}(N)\}_{N\geq 0}$ is

[TABLE]

Now, let181818with $y_{0}\succ y_{1}$. $Y_{0}=Y\cup\{y_{0}\}$ and let $Y_{0}^{*}$ denotes the free monoid generated by $Y_{0}$ admitting $1_{Y_{0}^{*}}$ as neutral element. As in (1), let us introduce another correspondence

[TABLE]

In all the sequel, for some convenience, we will also adopt the following notations, for any $w=y_{s_{1}}\ldots y_{s_{r}}\in Y_{0}^{*}$,

[TABLE]

Example 9 ($\operatorname{Li}^{-}_{y_{0}^{r}}$ and $\mathrm{H}^{-}_{y_{0}^{r}}$)

By Proposition (5), we have $\operatorname{Li}^{-}_{y_{0}^{r}}=\lambda^{r}$. Hence,

[TABLE]

Definition 4

With the convention $\mathrm{H}^{-}_{1_{Y^{*}_{0}}}=1$, we put

[TABLE]

Since, for $y_{k}\in Y,u\in Y^{*}$ (resp. $y_{k}\in Y_{0},u\in Y_{0}^{*}$) and $N\geq 1$, one has $\mathrm{H}_{y_{k}u}(N)-\mathrm{H}_{y_{k}u}(N-1)=N^{-k}\mathrm{H}_{u}(N-1)$ (resp. $\mathrm{H}^{-}_{y_{k}u}(N)-\mathrm{H}^{-}_{y_{k}u}(N-1)=N^{k}\mathrm{H}^{-}_{u}(N-1)$). Then

Proposition 3

$\mathrm{H}$* and $\mathrm{H}^{-}$ satisfy the following difference equations*

[TABLE]

Hence, for any $w\in Y^{*}$ (resp. $w\in Y_{0}^{*}$), $\mathrm{H}_{w}(N)$ (resp. $\mathrm{H}^{-}_{w}(N)$) is of valuation $N$.

In all the sequel, the length and the weight of $u=y_{i_{1}}\ldots y_{i_{k}}\in Y^{*}$ are defined respectively as the numbers $\mid\!u\!\mid=k$ and $(u)=i_{1}+\ldots+i_{k}$.

Definition 5

Let $g,h\in{\mathbb{Q}}\left\langle{\left\langle{Y_{0}}\right\rangle}\right\rangle[\![t]\!]$ be defined as follows (here, $\mid\!1_{Y_{0}^{*}}\!\mid=(1_{Y_{0}^{*}})=0$)

[TABLE]

Remark 1

The generating series $h$ is an extension of the Euler series $\sum_{n\geq 0}n!t^{n}$ and it can be obtained as Borel-Laplace transform of $g$. 2. 2.

The ordinary generating series ${\cal Y}(t):=1+\sum_{r\geq 0}y_{r}\;t^{r}$ and its inverse are group-like. The generating series $\Lambda(t)=\sum_{w\in Y^{*}_{0}}t^{(w)+|w|}w$ can be obtained from $1/{\cal Y}(t)$ by use the following change of alphabet $y_{r}\leftarrow ty_{r}$ it can be expressed as

[TABLE]

Now, let us consider the following differential and integration operators acting on ${\mathbb{C}}\{\operatorname{Li}_{w}\}_{w\in X^{*}}$ which can be extended over ${\mathcal{C}}\{\operatorname{Li}_{w}\}_{w\in X^{*}}$ orlando  :

[TABLE]

Let $\Theta$ and $\Im$ be monoid morphisms such that $\Theta(1_{X^{*}})=\Im(1_{X^{*}})=\mathrm{Id}$ and, for $x_{i}\in X,v\in X^{*}$, $\Theta(vx_{i})=\Theta(v)\theta_{i}$ and $\Im(vx_{i})=\Im(v)\iota_{i}$. By extension, we obtain ${\mathcal{H}}_{\tt conc}\cong({\mathbb{Q}}\langle\Theta(X)\rangle,{\tt conc},\mathrm{Id},\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}},\epsilon)$ and ${\mathcal{H}}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}\cong({\mathbb{Q}}\langle\Im(X)\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},\mathrm{Id},\Delta_{\tt conc},\epsilon)$. Hence,

Proposition 4

The operators $\{\theta_{0},\theta_{1},\iota_{0},\iota_{1}\}$ satisfy in particular,

[TABLE] 2. 2.

For any $w=y_{s_{1}}\ldots y_{s_{r}}\in Y^{*}$ ($\pi_{X}(w)=x_{0}^{s_{1}-1}x_{1}\ldots x_{0}^{s_{r}-1}x_{1}$) and $u=y_{t_{1}}\ldots y_{t_{r}}\in Y_{0}^{*}$, we can rephrase $\operatorname{Li}_{w},\operatorname{Li}^{-}_{u}$ as follows

[TABLE] 3. 3.

${\mathcal{C}}\{\operatorname{Li}_{w}\}_{w\in X^{*}}\cong{\mathcal{C}}\otimes{\mathbb{C}}\{\operatorname{Li}_{w}\}_{w\in X^{*}}$* is closed under of $\iota_{0},\iota_{1},\theta_{0},\theta_{1}$.* 4. 4.

Let ${\lambda(z):={z}/{(1-z)}}\in{\mathcal{C}}$. Then $\lambda$ and $1/{\lambda}$ are the eigenvalues of $\theta_{0}\iota_{1}$ and $\theta_{1}\iota_{0}$ within ${\mathcal{C}}\{\operatorname{Li}_{w}\}_{w\in X^{*}}$ respectively :

[TABLE] 5. 5.

For any $n\geq 0$ and $w\in X^{*}$, one has191919For any $w=x_{i_{1}}\ldots x_{i_{r}}\in X^{*}$, we denote $\widetilde{w}=x_{i_{r}}\ldots x_{i_{1}}$.**

[TABLE] 6. 6.

For any $P,Q\in{\mathbb{Q}}\langle X\rangle$ and $R\in{\cal L}ie_{{\mathbb{Q}}}\langle X\rangle$, one has

[TABLE]

Proof

The proofs are immediate.

Proposition 5

GHN **

For any $w\in Y_{0}^{*}$, one has $\operatorname{Li}^{-}_{w}(z)=\lambda^{\mid\!w\!\mid}(z){A^{-}_{w}(z)}{(1-z)^{-(w)}}$, where $A^{-}_{w}$ is the extended Eulerian polynomial defined recursively as follows

[TABLE]

and $A_{n,k}$ are Eulerian numbers satisfying $A_{n,k}=\sum_{j=0}^{k}(-1)^{j}{n+1\choose j}(k+1-j)^{n}$. 2. 2.

For any $w\in Y^{*}$, let us define $\{G^{-}_{w}(n)\}_{n\in{\mathbb{N}}}$ by the following generating series

[TABLE]

Then $\mathrm{H}^{-}_{w}(N)=(N+1)N(N-1)\ldots(N-\mid\!w\!\mid+1)G^{-}_{w}(N)$. 3. 3.

$\operatorname{Li}^{-}_{w}(z)\in{\mathbb{Q}}[(1-z)^{-1}]\subsetneq{\mathcal{C}}$* and $\mathrm{H}^{-}_{w}(N)\in{\mathbb{Q}}[N]$ of degree $\mid\!w\!\mid+(w)$.*

Example 10

GHN [Case of $r=1$ by Maple]

Since ${A_{n}(z)}/{(1-z)^{n+1}}=\sum_{j\geq 0}z^{j}(j+1)^{n}$ then $\operatorname{Li}^{-}_{y_{n}}(z)=zA_{n}(z)/(1-z)^{n+1}$ (see foata for example). For example,

[TABLE] 2. 2.

For any positive integer $m$, one has

[TABLE]

where $B_{k}$ is the $k$-th Bernoulli’s number given by its exponential generating series

[TABLE]

For example, (recall that $B_{0}=1,B_{1}=-1/2,B_{2}=1/6,B_{3}=0,B_{4}=-1/30$),

[TABLE]

Example 11

GHN [Case of $r=2$ by Maple]

From what precedes, $\operatorname{Li}^{-}_{y_{m}y_{n}}=(\theta_{0}^{m+1}\iota_{1})\operatorname{Li}^{-}_{y_{n}}=\theta_{0}^{m}(\theta_{0}\iota_{1})\operatorname{Li}^{-}_{y_{n}}$. Since, by Example 9, we have $(\theta_{0}\iota_{1})\operatorname{Li}^{-}_{y_{n}}=\operatorname{Li}^{-}_{y_{0}}\operatorname{Li}^{-}_{y_{n}}$ then $\operatorname{Li}^{-}_{y_{m}y_{n}}=\theta_{0}^{m}[\operatorname{Li}^{-}_{y_{0}}\operatorname{Li}^{-}_{y_{n}}]=\sum_{l=0}^{m}{m\choose l}\operatorname{Li}^{-}_{y_{l}}\operatorname{Li}^{-}_{y_{m+n-l}}$. For example,

[TABLE] 2. 2.

For any positive integers $m,n$, one has

[TABLE]

For example,

[TABLE]

Example 12

GHN [General case]

One has, for any $y_{s_{1}}u=y_{s_{1}}\ldots y_{s_{r}}\in Y^{*}_{0}$,

[TABLE]

Denoting $S_{2}({k_{i}},j)$ Stirling numbers of the second kind, one has

[TABLE]

In particular, if \mathop{\mathrm{missing}}{\omega}\nolimits\in Y^{*} then $(1-z)^{\mid\!w\!\mid}\operatorname{Li}^{-}_{w}(z)$ is polynomial of degree $(w)$ in $\lambda(z)$. 2. 2.

We define, firstly, the polynomials $\{B_{y_{n_{1}}\ldots y_{n_{r}}}(z)\}_{n_{1},\ldots,n_{r}\in{\mathbb{N}}}$ by their commutative exponential generating series as follows, for $z\in{\mathbb{C}}$,

[TABLE]

or by the difference equation, for $n_{1}\in{\mathbb{N}}_{+}$,

[TABLE]

For any $w\in y_{s}Y_{0}^{*},s>1$, we have $B_{w}(1)=B_{w}(0)$. Then let also, for any $1\leq k\leq r$,

[TABLE]

Then we have the extended Faulhaber’s identities

[TABLE]

Proposition 6

GHN * The following maps are morphisms of algebras*

[TABLE]

Proof

Recall that the quasi-symmetric functions on the variables ${\bf t}=\{t_{i}\}_{N\geq i\geq 1}$, i.e.

[TABLE]

satisfy the quasi-shuffle relation reutenauer , i.e. for any $u,v\in Y_{0}^{*}$, \mathrm{F}_{u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v}({\bf t})=\mathrm{F}_{u}({\bf t})\mathrm{F}_{v}({\bf t}).

Since $\mathrm{H}^{-}_{s_{1},\ldots,s_{r}}(N)$ can be obtained by specializing, in $\mathrm{F}_{s_{1},\ldots,s_{r}}({\bf t})$, the variables ${\bf t}$ at

[TABLE]

then $\mathrm{H}^{-}$ is a morphism of algebras. Therefore, $\mathrm{P}^{-}$ is also a morphism of algebras.

Global renormalizations via noncommutative generating series

By (2.3), $\mathrm{L}$ and $\mathrm{H}$ are images, by the tensor products $\operatorname{Li}\otimes\mathrm{Id}$ and $\mathrm{H}\otimes\mathrm{Id}$, of the diagonal series ${\cal D}_{X}$ and ${\cal D}_{Y}$ respectively. Then we get

Theorem 2.5 (Factorization of $\mathrm{L}$ and of $\mathrm{H}$, SLC43 ; FPSAC98 ; acta )

Let

[TABLE]

Then \mathrm{L}(z)=e^{-x_{1}\log(1-z)}\mathrm{L}_{\mathop{\mathrm{missing}}{reg}\nolimits}(z)e^{x_{0}\log z} and \mathrm{H}(N)=e^{\mathrm{H}_{y_{1}}(N)\;y_{1}}\mathrm{H}_{\mathop{\mathrm{missing}}{reg}\nolimits}(N).

For any $l\in{\mathcal{L}yn}X-X$ (resp. ${\mathcal{L}yn}Y-\{y_{1}\}$), the polynomial $S_{l}$ (resp. $\Sigma_{l}$) is a finite combination of words in $x_{0}X^{*}x_{1}$ (resp. $Y^{*}-y_{1}Y^{*}$). Then we can state

Proposition 7 (acta )

Let Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}:=\mathrm{L}_{\mathop{\mathrm{missing}}{reg}\nolimits}(1) and Z_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}:=\mathrm{H}_{\mathop{\mathrm{missing}}{reg}\nolimits}(\infty). Then $Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and Z_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} are group-like, for $\Delta_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and \Delta_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} respectively.

Proposition 8 (Successive integrations and differentiations of $\mathrm{L}$, orlando )

We have, for any $n\in{\mathbb{N}}$,

$\iota_{0}^{n}\mathrm{L}=x_{0}^{n}\triangleright\mathrm{L}$* and $\iota_{1}^{n}\mathrm{L}=x_{1}^{n}\triangleright\mathrm{L}$.* 2. 2.

$\partial_{z}^{n}\mathrm{L}=D_{n}\mathrm{L}$* and $\theta_{0}^{n}\mathrm{L}=E_{n}\mathrm{L}$, where202020Since $\theta_{0}+\theta_{1}=\partial_{z}$ then we also have $\theta_{1}^{n}\mathrm{L}(z)=[D_{n}(z)-E_{n}(z)]\mathrm{L}(z)$. The more general actions of $\{\Theta(w)\}_{w\in X^{*}}$ on $\mathrm{L}$ are more complicated to be expressed here. the polynomials $D_{n}$ and $E_{n}$ in ${\mathcal{C}}\langle X\rangle$ are*

[TABLE]

and for any $w=x_{i_{1}}\cdots x_{i_{k}}$ and ${\bf r}=(r_{1},\ldots,r_{k})$ of degree \mathop{\mathrm{missing}}{deg}\nolimits({\bf r})=k and of weight $\mathrm{wgt}({\bf r})=k+r_{1}+\cdots+r_{k}$, the polynomials $\tau_{\bf r}(w)=\tau_{r_{1}}(x_{i_{1}})\cdots\tau_{r_{k}}(x_{i_{k}})$ and $\rho_{\bf r}(w)=\rho_{r_{1}}(x_{i_{1}})\cdots\rho_{r_{k}}(x_{i_{k}})$ are defined respectively by, for any $r\in{\mathbb{N}}$,

[TABLE]

Example 13 (Coefficients of $\theta_{0}^{n}\mathrm{L}$)

Since, for any $u\in X^{+}$, $\theta_{0}\operatorname{Li}_{x_{0}u}=\operatorname{Li}_{u}$ and $\theta_{1}\operatorname{Li}_{x_{0}u}=\operatorname{Li}_{0}\operatorname{Li}_{u}$, one obtains for example

• •

For any $n\geq 1$ and $w\in X^{*}$, one has $\theta_{0}^{n}\operatorname{Li}_{x_{0}^{n}w}=\operatorname{Li}_{w}$. Hence,

[TABLE]

• •

$\theta_{0}\operatorname{Li}_{x_{1}^{2}}=\operatorname{Li}_{0}\operatorname{Li}_{x_{1}},\theta_{0}^{2}\operatorname{Li}_{x_{1}^{2}}=\operatorname{Li}^{-}_{\pi_{Y}(x_{1})}\operatorname{Li}_{x_{1}}+\operatorname{Li}_{0}^{2},\theta_{0}^{3}\operatorname{Li}_{x_{1}^{2}}=\operatorname{Li}^{-}_{\pi_{Y}(x_{0}x_{1})}\operatorname{Li}_{x_{1}}+3\operatorname{Li}^{-}_{\pi_{Y}(x_{1})}\operatorname{Li}_{0}$ because

[TABLE]

The noncommutative generating series $\mathrm{L}$ satisfies the differential equation

[TABLE]

with boundary condition

[TABLE]

This implies that $\mathrm{L}$ is the exponential of a Lie series FPSAC98 ; SLC43 . Hence orlando ,

[TABLE]

Theorem 2.6 (orlando )

Let $G,H$ be exponential solutions of (18). Then there exists a constant Lie series $C$ such that $G=He^{C}$. 2. 2.

Let $\mathrm{Gal}_{{\mathbb{C}}}(DE)$ be the differential Galois group associated to the Drinfel’d equation. Then $\mathrm{Gal}_{{\mathbb{C}}}(DE)=\{e^{C}\mid C\in{\cal L}ie_{{\mathbb{C}}}\langle\!\langle X\rangle\!\rangle\}$ and it contains the monodromy group defined by ${\mathcal{M}}_{0}\mathrm{L}=\mathrm{L}\exp(2\mathrm{i}\pi\mathfrak{m}_{0})$ and ${\mathcal{M}}_{1}\mathrm{L}=\mathrm{L}Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}^{-1}\exp(-2\mathrm{i}\pi x_{1})Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}=\mathrm{L}\exp(2\mathrm{i}\pi\mathfrak{m}_{1})$, where $\mathfrak{m}_{0}=x_{0},\mathfrak{m}_{1}=\prod_{l\in{\mathcal{L}yn}X-X}^{\searrow}\exp(-\zeta(S_{l})\operatorname{ad}_{P_{l}})(-x_{1})$.

Then let us put212121Here, the coefficient $\langle B(y_{1})\mid y_{1}^{k}\rangle$ corresponds to the Euler–Mac Laurin constant associated to $\langle\mathrm{Const}(N)\mid y_{1}^{k}\rangle$, i.e. the finite party of its asymptotic expansion in the scale of comparison $\{n^{a}\log^{b}(n)\}_{a\in{\mathbb{Z}},b\in{\mathbb{N}}}$. $\Lambda:=\pi_{Y}\mathrm{L}$ and cade

[TABLE]

and finally, $B^{\prime}(y_{1}):=\exp(\gamma y_{1})B(y_{1})$. Hence, we get $\pi_{Y}\mathrm{P}(z)\;{}_{\widetilde{z\rightarrow 1}}\;\mathrm{Mono}(z)\pi_{Y}Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and $\mathrm{H}(N)\;{}_{\widetilde{N\rightarrow+\infty}}\;\mathrm{Const}(N)\pi_{Y}Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ as a consequence of (20)-(21). Or equivalently,

Theorem 2.7 (First global renormalizations of divergent polyzetas, cade )

[TABLE]

Theorem 2.8 (AofA )

For any $g\in{\mathcal{C}}\{\mathrm{P}_{w}\}_{w\in Y^{*}}$, there exist algorithmically computable coefficients $c_{j},b_{i}\in{\mathbb{C}},\alpha_{j},\eta_{i}\in{\mathbb{Z}},\beta_{j},\kappa_{i}\in{\mathbb{N}}$ such that

[TABLE]

Theorem 2.8 means also that the $\{\mathrm{P}_{w}\}_{w\in Y^{*}}$ admit a full singular expansion, at $1$, and then their ordinary Taylor coefficients, $\{\mathrm{H}_{w}\}_{w\in Y^{*}}$ admit a full asymptotic expansion, for $+\infty$. More precisely,

Corollary 1

For any $w\in X^{*}$ and for any $k,i,j\in{\mathbb{N}}$, $k\geq 1$, there exists uniquely determined coefficients $a_{i},b_{i,j}$ belonging to ${\mathcal{Z}}$; $\gamma_{\pi_{Y}(w)}$, $\alpha_{i}$ and $\beta_{i,j}$ belonging to the ${\mathbb{Q}}[\gamma]$-algebra generated by convergent polyzetas such that,

[TABLE]

and, likely

[TABLE]

Remark 2

i) The two expansions (23) and (24) are asymptotic expansions of $Li_{w}$ and $H_{w}$ with respect to the scales $(1-z)^{n}log(1-z)^{m};\ n,m\geq 0$ and $N^{-k}log(N)^{m};\ k,m\geq 0$ respectively.

ii) In (eq. 23), the error term $\mathrm{o}_{k}^{(1)}((1-z)^{k})$ can be put to the form $\mathrm{O}_{k}^{(1)}((1-z)^{k+\epsilon})$ for any $\epsilon\in]0,1[$.

More generally, by Theorem 2.6, we get

Proposition 9

For any commutative ${\mathbb{Q}}$-algebra $A$ and for any Lie series $C\in{\cal L}ie_{A}\langle X\rangle$, we set $\overline{\mathrm{L}}=\mathrm{L}e^{C},\overline{\Lambda}=\pi_{Y}\overline{\mathrm{L}}$ and $\overline{\mathrm{P}}(z)=(1-z)^{-1}{\overline{\Lambda}(z)}$, then

$\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}=Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}e^{C}$* is group-like, for the co-product $\Delta_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$,* 2. 2.

$\overline{\mathrm{L}}(z)\;{}_{\widetilde{z\rightarrow 1}}\;\exp(-x_{1}\log(1-z))\;\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$, 3. 3.

$\overline{\mathrm{P}}(z)\;{}_{\widetilde{z\rightarrow 1}}\;\mathrm{Mono}(z)\pi_{Y}\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$, 4. 4.

$\overline{\mathrm{H}}(N)\;{}_{\widetilde{N\rightarrow\infty}}\;\mathrm{Const}(N)\pi_{Y}\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$,

where, for any $w\in Y^{*}$ and $N\geq 0$, one defines the coefficient $\langle\overline{\mathrm{H}}(N)\mid w\rangle$ of $w$ in the power series $\overline{\mathrm{H}}(N)$ as the coefficient $\langle\overline{\mathrm{P}}_{w}(z)\mid z^{N}\rangle$ of $z^{N}$ in the ordinary Taylor expansion of the polylogarithmic function $\overline{\mathrm{P}}_{w}(z)$.

By Proposition 9, we get successively

Proposition 10 (SLC44 )

Let $\overline{\zeta}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and \overline{\zeta}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} be the characters of respectively $(A\langle X\rangle,{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}})$ and (A\langle Y\rangle,{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}) satisfying $\overline{\zeta}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}(x_{0})=\overline{\zeta}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}(x_{1})=0$ and \overline{\zeta}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}(y_{1})=0. Then

[TABLE]

Proposition 11

Let $\{\overline{\gamma}_{w}\}_{w\in Y^{*}}$ be the Euler–Mac Laurin constants associated to $\{\overline{\mathrm{H}}_{w}(N)\}_{w\in Y^{*}}$. Let $\overline{Z}_{\gamma}$ be the noncommutative generating series of these constants. Then,

The following map realizes a character :

[TABLE] 2. 2.

The noncommutative power series $\overline{Z}_{\gamma}$ is group-like, for \Delta_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}. 3. 3.

There exists a group-like element \overline{Z}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}, for the co-product \Delta_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}}, such that

[TABLE]

By Theorem 2.7, Propositions 9 and 11, we also get

Proposition 12

For any $C\in{\cal L}ie_{A}\langle X\rangle$ such that $\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}=Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}e^{C}$. Then

[TABLE]

where $B(y_{1})$ and $B^{\prime}(y_{1})$ are given in (22).

By Proposition 9, the noncommutative generating series $\overline{Z}_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and \overline{Z}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} are group-like, for the co-product $\Delta_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ and \Delta_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}} respectively, and we also have

[TABLE]

Hence, by Proposition 12, we deduce in particular,

[TABLE]

The elements of $\{\pi_{Y}P_{l}\}_{l\in{\mathcal{L}yn}X}$ are decomposable in the linear basis $\{\Pi_{w}\}_{w\in Y^{*}}$ of {\cal U}(\mathrm{Prim}({\mathcal{H}}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}})). Thus, by identification of local coordinates, i.e. the coefficients of $\{\Pi_{l}\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$ in the basis $\{\Sigma_{l}\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$, we get homogenous polynomial relations on polyzetas encoded by $\{\Sigma_{l}\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$ acta .

Proposition 13

There exist $A,B$ and $C\in{\mathbb{Q}}\left\langle{Y_{0}}\right\rangle$ such that

[TABLE]

where the series $g,h$ were defined in the definition 5.

Proof

By Propositions 5, for $w=y_{s_{1}}\ldots y_{s_{r}}$, there exists $a,b,c\in{\mathbb{Q}}$ such that

[TABLE]

Putting $\langle A\mid w\rangle=(-1)^{\mid\!w\!\mid}a,\langle B\mid w\rangle=(-1)^{\mid\!w\!\mid}b,\langle C\mid w\rangle=(-1)^{\mid\!w\!\mid}c$, it follows the expected results.

Proposition 14

GHN * For any $w\in Y_{0}^{*}$, there are non-zero constants, namely $C^{-}_{w}$ and $B^{-}_{w}$, which only depend on $w$ and $r$ such that*

[TABLE]

Moreover, $C^{-}_{w}$ and $B^{-}_{w}$ are well determined by

[TABLE]

Example 14

GHN [of $C^{-}_{w}$ and $B^{-}_{w}$]

[TABLE]

Proposition 15

GHN * Let $u,v\in Y_{0}^{*}$. We get \mathrm{H}^{-}_{u}\mathrm{H}^{-}_{v}=\mathrm{H}^{-}_{u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v}.*

Proof

Let $w\in Y_{0}^{*}$ associated to $s=(s_{1},\ldots,s_{k})$. The quasi-symmetric monomial functions on the commutative alphabet $t=\{t_{i}\}_{i\geq 1}$ are defined as follows

[TABLE]

For any $u,v\in Y_{0}^{*}$, we have M_{u}(t)M_{v}(t)=M_{u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v}(t). Then, the harmonic sum $\mathrm{H}^{-}_{s_{1},\ldots,s_{k}}(N)$ is obtained by specializing the indeterminates $t=\{t_{i}\}_{i\geq 1}$ from $M_{w}(t)$ as follows: $t_{i}=i\mbox{ for }1\leq i\leq N$ and $t_{i}=0\mbox{ for }N<i.$

Theorem 2.9 (Second global renormalizations of divergent polyzetas)

The generating series $\mathrm{H}^{-}$ is group-like and $\log\mathrm{H}^{-}$ is primitive. Moreover222222Here, the Hadamard product is denoted by $\odot$ and and its dual law, the diagonal comultiplication is denoted by $\Delta_{\odot}$. The series $g,h$ are defined in Definition 5.,

[TABLE] 2. 2.

$\ker\mathrm{H}^{-}_{\bullet}$* is a prime ideal of ({\mathbb{Q}}\left\langle{Y_{0}}\right\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;), i.e. ${\mathbb{Q}}\langle Y_{0}\rangle\setminus\ker\mathrm{H}^{-}_{\bullet}$ is closed by  ** **. *

Proof

The first result is a consequence of the extended Friedrichs criterion BDHHT ; acta ; VJM and the second is a consequence of Proposition 13.

Definition 6

For any $n\in{\mathbb{N}}_{+}$, let $\mathbb{P}_{n}:=\mathrm{span}_{{\mathbb{R}}_{+}}\{w\in Y_{0}^{*}|(w)+|w|=n\}\setminus\{0\}$ be the blunt232323i.e. without zero or see Appendix A. convex cone generated by the set $\{w\in Y_{0}^{*}|(w)+|w|=n\}$.

By definition, $C^{-}_{\bullet}$ is linear on the set $\mathbb{P}_{n}$. For any $u,v\in Y^{*}_{0}$, one has u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v=u\mathop{{}_{{}^{\sqcup\!\sqcup}}}v+\sum_{|w|<|u|+|v|\atop(w)=(u)+(v)}x_{w}w and the $x_{w}$’s are positive. Moreover, for any $w$ which belongs to the support of $\sum_{|w|<|u|+|v|\atop(w)=(u)+(v)}x_{w}w$, one has $(w)+|w|<(u)+(v)+|u|+|v|$, thus, by the definition of $C^{-}_{\bullet}$, one obtains

Corollary 2

Let $w,v\in Y^{*}_{0}$. Then C^{-}_{w}C^{-}_{v}=C^{-}_{w\mathop{{}_{{}^{\sqcup\!\sqcup}}}v}=C^{-}_{w\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v}. 2. 2.

For any $P,Q\notin\ker\mathrm{H}^{-}_{\bullet}$, C^{-}_{P}C^{-}_{Q}=C^{-}_{P\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;Q} and ${\mathbb{Q}}\left\langle{Y_{0}}\right\rangle\setminus\ker\mathrm{H}^{-}_{\bullet}$ is a \;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;- multiplicative monoid containing $Y^{*}_{0}$.

Now, let us prove that $C^{-}_{\bullet}$ can be extended as a character, for $\mathop{{}_{{}^{\sqcup\!\sqcup}}}$, or equivalently, $C^{-}$ is group-like (see the Freidrichs’ criterion reutenauer ) and then $\log{C}^{-}$ is primitive.

Lemma 2

Let $\mathcal{A}$ be an unitary ${\mathbb{R}}$-associative algebra and $f:\sqcup_{n\geq 0}\mathbb{P}_{n}\longrightarrow\mathcal{A}$ such that

For any $u,v\in Y^{*}_{0}$, $f(u\mathop{{}_{{}^{\sqcup\!\sqcup}}}v)=f(u)f(v)$. In particular, $f(1_{Y^{*}_{0}})=1_{\mathcal{A}}$. 2. 2.

On every $\mathbb{P}_{n}$, one has $f(\sum_{i\in I}\alpha_{i}w_{i})=\sum_{i\in I}\alpha_{i}f(w_{i})$, where $\alpha_{i}\in{\mathbb{R}}^{*}_{+}$.

Then $f$ can be extended uniquely as a character, i.e. $S_{f}=\sum_{w\in Y^{*}_{0}}f(w)w$ is group-like for $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$.

Proof

By definition of $f$ and $S_{f}$, it is immediate $\langle S_{f}\mid 1_{Y^{*}_{0}}\rangle=1_{\mathcal{A}}$. One can check easily that $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(S_{f})=S_{f}\otimes S_{f}$. Hence, $S_{f}$ is group-like, for $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$.

Corollary 3

The noncommutative generating series $C^{-}$ is group-like, for $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$.

Proof

It is a consequence of Lemma 2 and Corollary 2.

Example 15

GHN [of $C^{-}_{u\mathop{{}_{{}^{\sqcup\!\sqcup}}}v}$ and C^{-}_{u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v}] Let $Y_{0}=\{y_{i}\}_{i\geq 0}$ be an infinite alphabet.

[TABLE]

[TABLE]

In the above tables, it is clearly seen that $C^{-}_{\bullet}$ is linear on $\mathbb{P}_{n}$. For example, let $u=y_{1}$ and $v=y_{2}y_{5}$. Then $u\mathop{{}_{{}^{\sqcup\!\sqcup}}}v=y_{1}y_{2}y_{5}+y_{2}y_{1}y_{5}+y_{2}y_{5}y_{1}$. Hence, we get $C^{-}_{y_{1}y_{2}y_{5}}+C^{-}_{y_{2}y_{1}y_{5}}+C^{-}_{y_{2}y_{5}y_{1}}=\frac{1}{594}+\frac{1}{528}+\frac{1}{176}=\frac{1}{108}=C^{-}_{y_{1}}C^{-}_{y_{2}y_{5}}=C^{-}_{y_{1}\mathop{{}_{{}^{\sqcup\!\sqcup}}}y_{2}y_{5}}$. Note that $y_{1}y_{2}y_{5},y_{2}y_{1}y_{5},y_{2}y_{5}y_{1}\in\mathbb{P}_{11}$. But we have also u\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;v=y_{1}y_{2}y_{5}+y_{2}y_{1}y_{5}+y_{2}y_{5}y_{1}+y_{3}y_{5}+y_{2}y_{6}. Moreover, $C^{-}_{y_{1}y_{2}y_{5}}+C^{-}_{y_{2}y_{1}y_{5}}+C^{-}_{y_{2}y_{5}y_{1}}+C^{-}_{y_{3}y_{5}}+C^{-}_{y_{2}y_{6}}=\frac{1}{108}+\frac{13}{420}\neq\frac{1}{108}=C^{-}_{y_{1}}C^{-}_{y_{2}y_{5}}.$ However, from $y_{3}y_{5},y_{2}y_{6}\in\mathbb{P}_{10}$, we can conclude that

[TABLE]

3 Polysystems and differential realization

3.1 Polysystems and convergence criterion

Estimates (from above) for series

Here, $({\mathbb{K}},\|.\|)$ is a normed space.

Definition 7

(these ; cade ) Let $\xi,\chi$ be real positive functions over $X^{*}$. Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$.

$S$ will be said $\xi-$* exponentially bounded from above* if it satisfies

[TABLE]

We denote by ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$ the set of formal power series in ${\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ which are $\xi-$ exponentially bounded from above. 2. 2.

$S$ satisfies the $\chi-$growth condition if it satisfies

[TABLE]

We denote by ${\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$ the set of formal power series in ${\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ satisfying the $\chi-$growth condition.

Lemma 3

If $R=\displaystyle\sum_{w\in X^{*}}\mid\!w\!\mid!\;w$ then \langle R^{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}2}\mid w\rangle=\displaystyle\sum_{u,v\in X^{*}\atop\mathop{\mathrm{missing}}{supp}\nolimits(u\mathop{{}_{{}^{\sqcup\!\sqcup}}}v)\ni w}\mid\!u\!\mid!\mid\!v\!\mid!\leq 2^{\mid\!w\!\mid}\mid\!w\!\mid!.

Proof

One has

[TABLE]

The last sum is equal to $(1+\mid\!w\!\mid)\mid\!w\!\mid!$. By induction on $|w|$, one has $1+|w|\leq 2^{|w|}$. Then the expected result follows.

Proposition 16

If $S_{1},S_{2}$ satisfy the growth condition then $S_{1}+S_{2},S_{1}\mathop{{}_{{}^{\sqcup\!\sqcup}}}S_{2}$ do also.

Proof

It is immediate for $S_{1}+S_{2}$. Next, since $\|\langle S_{i}\mid w\rangle\|\leq K_{i}\chi_{i}(w)\mid\!w\!\mid!$ then

[TABLE]

Let $K=K_{1}K_{2}$ and let $\chi$ be a real positive function over $X^{*}$ such that, for any $w\in X^{*}$

[TABLE]

With the notations in Lemma 3, we get $\|\langle S_{1}\mathop{{}_{{}^{\sqcup\!\sqcup}}}S_{2}\mid w\rangle\|\leq K\chi(w)\langle S_{1}R^{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}2}\mid w\rangle$. Hence, $S_{1}\mathop{{}_{{}^{\sqcup\!\sqcup}}}S_{2}$ satisfies the $\chi^{\prime}$-growth condition with $\chi^{\prime}(w)=2^{\mid\!w\!\mid}\chi(w).$

Definition 8

(these ; cade ) Let $\xi$ be a real positive function defined over $X^{*}$, $S$ will be said $\xi$-exponentially continuous if it is continuous over ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$. The set of formal power series which are $\xi$-exponentially continuous is denoted by ${\mathbb{K}}^{\xi-ec}\langle\!\langle X\rangle\!\rangle.$

Lemma 4

these ; cade * For any real positive function $\xi$ defined over $X^{*}$, we have ${\mathbb{K}}\langle X\rangle\subset{\mathbb{K}}^{\xi-ec}\langle\!\langle X\rangle\!\rangle$. Otherwise, for $\xi=0$, we get ${\mathbb{K}}\langle X\rangle={\mathbb{K}}^{0-ec}\langle\!\langle X\rangle\!\rangle$. Hence, any polynomial is $0-$exponentially continuous.*

Proposition 17 (these ; cade )

Let $\xi,\chi$ be real positive functions over $X^{*}$ and $P\in{\mathbb{K}}\langle X\rangle$.

Let $S\in{\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$. The right residual of $S$ by $P$ belongs to ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$. 2. 2.

Let $R\in{\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$. The concatenation $SR$ belongs to ${\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$. 3. 3.

Moreover, if $\xi$ and $\chi$ are morphisms over $X^{*}$ satisfying $\sum_{x\in X}\chi(x)\xi(x)<1$ then, for any $F\in{\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$, $F$ is continuous over ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$.

Proof

Since $S\in{\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$ then

[TABLE]

If u\in\mathop{\mathrm{missing}}{supp}\nolimits(P) then, for any $w\in X^{*}$, one has $\langle S\triangleright u\mid w\rangle=\langle S\mid uw\rangle$ and $S\triangleright u$ belongs to ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$:

[TABLE]

It follows that $S\triangleright P$ is ${\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$ by taking K_{1}=K\max_{u\in\mathop{\mathrm{missing}}{supp}\nolimits(P)}\xi(u). 2. 2.

Since $R\in{\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$ then

[TABLE]

Let v\in\mathop{\mathrm{missing}}{supp}\nolimits(P) such that $v\neq\epsilon$. Since $Rv$ belongs to ${\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$ and one has, for $w\in X^{*}$, $\langle Rv\mid w\rangle=\langle R\mid v\triangleleft w\rangle$, i.e. there exists $K\in{\mathbb{R}}_{+},n\in{\mathbb{N}}$ such that

[TABLE]

Note if $v\triangleleft w=0$ then $\langle Rv\mid w\rangle=0$ and the previous conclusion holds. It follows that $RP$ is ${\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle$ by taking K_{2}=K\min_{v\in\mathop{\mathrm{missing}}{supp}\nolimits(P)}\chi(v)^{-1}. 3. 3.

Let $\xi,\chi$ be functions which satisfy the upper bound condition. The following quantity is well defined

[TABLE]

If $F\in{\mathbb{K}}^{\chi-\mathrm{gc}}\langle\!\langle X\rangle\!\rangle,C\in{\mathbb{K}}^{\xi-\mathrm{em}}\langle\!\langle X\rangle\!\rangle$ then there exist $K_{i}\in{\mathbb{R}}_{+},n_{i}\in{\mathbb{N}},i=1,2$ such that, for $w\in X^{\geq n_{i}}$, $\|\langle F\mid w\rangle\|\leq K_{1}\chi(w)\mid\!w\!\mid!$ and $\|\langle C\mid w\rangle\|\leq K_{2}{\xi(w)}/{\mid\!w\!\mid!}$. Thus,

[TABLE]

Upper bounds à la Cauchy

The algebra of formal power series on commutative indeterminates $\{q_{1},\ldots,q_{n}\}$ with coefficients in ${\mathbb{C}}$ is denoted by ${\mathbb{C}}[\![q_{1},\ldots,q_{n}]\!]$.

Definition 9

(these ; cade ) Let $f=\in{\mathbb{C}}[\![q_{1},\ldots,q_{n}]\!]$. We set

[TABLE]

$f$ is convergent if ${\rm CV}(f)\neq\emptyset$. Let ${\cal U}\subset{\mathbb{C}}^{n}$ be an open domain and $q\in{\mathbb{C}}^{n}$. $f$ is convergent on $q$ (resp. over ${\cal U}$) if $q\in{\rm CV}(f)$ (resp. ${\cal U}\subset{\rm CV}(f)$). We set ${\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]:=\{f\in{\mathbb{C}}[\![q_{1},\ldots,q_{n}]\!]:{\rm CV}(f)\neq\emptyset\}$. Let $q\in{\rm CV}(f)$. There exist $C_{f}\in{\mathbb{R}}_{+},\rho\in E(f),{\bar{\rho}}\in{\breve{E}}\!(f)$ such that $\mid\!q_{1}\!\mid<{\bar{\rho}}_{1}<\rho_{1},\ldots,\mid\!q_{n}\!\mid<{\bar{\rho}}_{n}<\rho_{n}$ and $\mid\!f_{i_{1},\ldots,i_{n}}\!\mid\rho_{1}^{i_{1}}\ldots\rho_{n}^{i_{n}}\leq C_{f}$, for $i_{1},\ldots,i_{n}\geq 0.$

The convergence modulus of $f$ at $q$ is $(C_{f},\rho,{\bar{\rho}})$.

Suppose ${\rm CV}(f)\neq\emptyset$ and let $q\in{\rm CV}(f)$. If $(C_{f},\rho,{\bar{\rho}})$ is a convergence modulus of $f$ at $q$ then $\mid\!f_{i_{1},\ldots,i_{n}}q_{1}^{i_{1}}\ldots q_{n}^{i_{n}}\!\mid\leq C_{f}({\bar{\rho}_{1}}/{\rho_{1}})^{i_{1}}\ldots({\bar{\rho}_{1}}/{\rho_{1}})^{i_{n}}$. Hence, at $q$, $f$ is majored termwise by $C_{f}\prod_{k=0}^{m}(1-{\bar{\rho}_{k}}/{\rho_{k}})^{-1}$ and it is uniformly absolutely convergent in $\{q\in{\mathbb{C}}^{n}:\mid\!q_{1}\!\mid<\;{\bar{\rho}},\ldots,\mid\!q_{n}\!\mid<{\bar{\rho}}\}$ which is open in ${\mathbb{C}}^{n}$. Thus, ${\rm CV}(f)$ is open in ${\mathbb{C}}^{n}$. Since the partial derivation $D^{j_{1}}_{1}\ldots D^{j_{n}}_{n}f$ is estimated by

[TABLE]

Proposition 18

(these )* We have ${\rm CV}(f)\subset{\rm CV}(D^{j_{1}}_{1}\ldots D^{j_{n}}_{n}f)$.*

Let $f\in{\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$. Let $\{A_{i}\}_{i=0,1}$ be a polysystem defined as follows

[TABLE]

Let $(\rho,\bar{\rho},C_{f}),\{(\rho,\bar{\rho},C_{i})\}_{i=0,1}$ be convergence modulus at $q\in{\rm CV}(f)\cap_{i=0,1,j=1,\ldots,n}{\rm CV}(A_{i}^{j})$ of $f$ and $\{A_{i}^{j}\}_{j=1,\ldots,n}$. Let us consider the following monoid morphisms

[TABLE]

Lemma 5

(fliessrealisation )* For $i=0,1$ and $j=1,\ldots,n$, one has $A_{i}\circ q_{j}=A_{i}^{j}$. Hence,*

[TABLE]

Lemma 6

(fliess1 )* For any word $w$, ${\cal A}(w)$ is continuous over ${\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$ and, for any $f,g\in{\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$, one has*

[TABLE]

These notations are extended, by linearity, to ${\mathbb{K}}\langle X\rangle$ and we will denote ${\cal A}(w)\circ f_{|q}$ the evaluation of ${\cal A}(w)\circ f$ at $q$.

Definition 10

(fliess1 ) Let $f\in{\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$. The generating series of the polysystem $\{A_{i}\}_{i=0,1}$ and of the observation $f$ is given by

[TABLE]

Then the following generating series is called Fliess generating series of the polysystem $\{A_{i}\}_{i=0,1}$ and of the observation $f$ at $q$:

[TABLE]

Lemma 7

(fliess1 )* The map $\sigma:({\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!],.)\longrightarrow({\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]\langle\!\langle X\rangle\!\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ is an algebra morphism, i.e. for any $f,g\in{\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$ and $\mu,\nu\in{\mathbb{C}}$, one has $\sigma(\nu f+\mu h)=\nu\sigma f+\mu\sigma g$ and $\sigma(fg)=\sigma f\mathop{{}_{{}^{\sqcup\!\sqcup}}}\sigma g$.*

Lemma 8

(fliessrealisation )* For any $w\in X^{*}$, $\sigma({\cal A}(w)\circ f)=w\triangleleft\sigma f\in{\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]\langle\!\langle X\rangle\!\rangle$.*

Theorem 3.1

(these )**

Let $\tau=\min_{1\leq k\leq n}\rho_{k}$ and $r=\max_{1\leq k\leq n}{\bar{\rho}_{k}}/{\rho_{k}}$. We have

[TABLE] 2. 2.

Let $K=C_{f}{(n+1)}{(1-r)^{-n}}$ and $\chi$ be the real positive function defined over $X^{*}$ :

[TABLE]

Then242424It is the same for the Fliess generating series $\sigma f_{|q}$ of $\{A_{i}\}_{i=0,1}$ and of $f$ at $q$.* the generating series $\sigma f$ of the polysystem $\{A_{i}\}_{i=0,1}$ and of the observation $f$ satisfies the *$\chi-$growth condition.

3.2 Polysystem and nonlinear differential equation

Nonlinear differential equation (with three singularities)

Let us consider the singular inputs252525These singular inputs are not included in the studies of Fliess motivated, in particular, by the renormalization of $y$ at $+\infty$ fliess1 ; fliessrealisation . $u_{0}(z):=z^{-1}$ and $u_{1}(z):=(1-z)^{-1}$, and

[TABLE]

where the state $q=(q_{1},\ldots,q_{n})$ belongs to a complex analytic manifold of dimension $n$, $q_{0}$ is the initial state, the observation $f$ belongs to ${\mathbb{C}}^{\rm cv}[\![q_{1},\ldots,q_{n}]\!]$ and $\{A_{i}\}_{i=0,1}$ is the polysystem defined on (25).

Definition 11

(hoangjacoboussous ) The following power series is called transport operator262626It plays the rôle of the resolvent in Mathematics and the evolution operator in Physics. of the polysystem $\{A_{i}\}_{i=0,1}$ and of the observation $f$

[TABLE]

By the factorization of the monoid by Lyndon words, we have hoangjacoboussous

[TABLE]

The Chen generating series along the path $z_{0}\rightsquigarrow z$, associated to $\omega_{0},\omega_{1}$ is

[TABLE]

which solves the differential equation (18) with the initial condition $S_{z_{0}\rightsquigarrow z_{0}}=1$. Thus, $S_{z_{0}\rightsquigarrow z}$ and $\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}$ satisfy the same differential equation taking the same value at $z_{0}$ and $S_{z_{0}\rightsquigarrow z}=\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}$. Any Chen generating series $S_{z_{0}\rightsquigarrow z}$ is group like ree and depends only on the homotopy class of $z_{0}\rightsquigarrow z$ chen . The product of $S_{z_{1}\rightsquigarrow z_{2}}$ and $S_{z_{0}\rightsquigarrow z_{1}}$ is $S_{z_{0}\rightsquigarrow z_{2}}=S_{z_{1}\rightsquigarrow z_{2}}S_{z_{0}\rightsquigarrow z_{1}}$. Let ${\varepsilon}\in]0,1[$ and $z_{i}={\varepsilon}\exp({\mathrm{i}}\beta_{i})$, for $i=0,1$. We set $\beta=\beta_{1}-\beta_{0}$. Let $\Gamma_{0}({\varepsilon},\beta_{0})$ (resp. $\Gamma_{1}({\varepsilon},\beta_{1})$) be the path turning around [math] (resp. $1$) in the positive direction from $z_{0}$ to $z_{1}$. By induction on the length of $w$, one has $\mid\!\langle S_{\Gamma_{i}({\varepsilon},\beta)}\mid w\rangle\!\mid=(2{\varepsilon})^{\mid\!w\!\mid_{x_{i}}}{\beta^{\mid\!w\!\mid}}/{\mid\!w\!\mid!}$, where $\mid\!w\!\mid$ denotes the length of $w$ and $\mid\!w\!\mid_{x_{i}}$ denotes the number of occurrences of letter $x_{i}$ in $w$, for $i=0$ or $1$. When ${\varepsilon}$ tends to $0^{+}$, these estimations yield $S_{\Gamma_{i}({\varepsilon},\beta)}=e^{{\mathrm{i}}\beta x_{i}}+o({\varepsilon})$. In particular, if $\Gamma_{0}({\varepsilon})$ (resp. $\Gamma_{1}({\varepsilon})$) is a circular path of radius ${\varepsilon}$ turning around [math] (resp. $1$) in the positive direction, starting at $z={\varepsilon}$ (resp. $1-{\varepsilon}$), then, by the noncommutative residue theorem FPSAC98 ; SLC43 , we get

[TABLE]

Finally, the asymptotic behaviors of $\mathrm{L}$ on (19) give SLC43 ; FPSAC98

[TABLE]

In other terms, $Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ is the regularized Chen generating series $S_{{\varepsilon}\rightsquigarrow 1-{\varepsilon}}$ of diffferential forms $\omega_{0}$ and $\omega_{1}$: $Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}$ is the noncommutative generating series of the finite parts of the coefficients of the Chen generating series $e^{x_{1}\log{\varepsilon}}\;S_{{\varepsilon}\rightsquigarrow 1-{\varepsilon}}\;e^{x_{0}\log{\varepsilon}}$.

Asymptotic behavior via extended Fliess fundamental formula

Theorem 3.2

(cade )* $y(z)=\mathcal{T}\circ f_{|_{q_{0}}}=\langle\sigma f_{|_{q_{0}}}\mid S_{z_{0}\rightsquigarrow z}\rangle$.*

This extends then Fliess’ fundamental formula fliess1 . By Theorem 2.5, the expansions of the output $y$ of nonlinear dynamical system with singular inputs follow

Corollary 4 (Combinatorics of Dyson series)

[TABLE]

where, for any word $w$ in $X^{*},g_{w}$ belongs to the polylogarithm algebra.

Since $S_{z_{0}\rightsquigarrow z}=\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}$ and $\sigma f_{|_{q_{0}}},\mathrm{L}(z_{0})^{-1}$ are invariant by $\partial_{z}=d/dz$ and $\theta_{0}=zd/dz$ then we get the $n$-th order differentiation of $y$, with respect to $\partial_{z}$ and $\theta_{0}$ :

[TABLE]

With the notations of Proposition 8, we get respectively

[TABLE]

For $z_{0}={\varepsilon}\rightarrow 0^{+}$, the asymptotic behavior and the renormalization at $z=1$ of $\partial_{z}^{n}y$ and $\theta_{0}^{n}y$ (or the asymptotic expansion and the renormalization of its Taylor coefficients at $+\infty$) are deduced from (34) and extend a little bit results of cade ; acta  :

Corollary 5 (Asymptotic behavior of output)

The $n$-order differentiation of the output $y$ of the system (31) is a ${\mathcal{C}}$-combination of the elements $g$ belonging to the polylogarithm algebra and272727Moreover, we get more out of this i.e. $\theta_{1}^{n}y(z)=\langle\sigma f_{|_{q_{0}}}\mid\!\mid\theta_{1}^{n}S_{z_{0}\rightsquigarrow z}\rangle=\langle\sigma f_{|_{q_{0}}}\mid\!\mid\theta_{1}^{n}\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}\rangle$. Therefore, $\theta_{1}^{n}y(z)=\langle\sigma f_{|_{q_{0}}}\mid[D_{n}(z)-E_{n}(z)]\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}\rangle=\langle\sigma f_{|_{q_{0}}}\triangleright[D_{n}(z)-E_{n}(z)]\mid\mathrm{L}(z)\mathrm{L}(z_{0})^{-1}\rangle$. Hence, $\theta_{1}^{n}y(1)\;{}_{\widetilde{{\varepsilon}\rightarrow 0^{+}}}\;\sum_{w\in X^{*}}\langle{\cal A}(w)\circ f_{|_{q_{0}}}\mid w\rangle\langle[D_{n}(1-{\varepsilon})-E_{n}(1-{\varepsilon})]e^{-x_{1}\log{\varepsilon}}\;Z_{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}}\;e^{-x_{0}\log{\varepsilon}}\mid w\rangle$. The actions of $\theta_{0}=u_{0}(z)^{-1}d/dz$ and $\theta_{1}=u_{1}(z)^{-1}d/dz$ over $y$ are equivalent to those of the residuals of $\sigma f_{|_{q_{0}}}$ by respectively $x_{0}$ and $x_{1}$. They correspond to functional differentiations causaldifferentiation while $\partial_{z}=d/dz$ is the ordinary differentiation and is equivalent to the residual by $x_{0}+x_{1}$., for any $n\geq 0$,

[TABLE] 2. 2.

If the ordinary Taylor expansions of $\partial_{z}^{n}y$ and $\theta_{0}^{n}y$ exist then the coefficients of these expansions belong to the algebra of harmonic sums and there exist algorithmically computable coefficients $a_{i},a^{\prime}_{i}\in{\mathbb{Z}},b_{i},b^{\prime}_{i}\in{\mathbb{N}},c_{i},c^{\prime}_{i}\in{\mathcal{Z}}[\gamma]$ such that

[TABLE]

3.3 Differential realization

Differential realization

Definition 12

The Lie rank of a formal power series $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ is the dimension of the vector space generated by

[TABLE]

Definition 13

Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ and let us put $\mathrm{Ann}(S):=\{\Pi\in{\cal L}ie_{{\mathbb{K}}}\langle X\rangle\mid S\triangleright\Pi=0\}$, and $\mathrm{Ann}^{\bot}(S):=\{Q\in({\mathbb{K}}\langle\!\langle X\rangle\!\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}})\mid Q\triangleright\mathrm{Ann}(S)=0\}$.

It is immediate that $\mathrm{Ann}^{\bot}(S)\ni S$. It follows then( see fliessrealisation ; reutenauerrealisation and Lemma 7),

Lemma 9

Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$. Then

If $S$ is of finite Lie rank, $d$, then the dimension of $\mathrm{Ann}^{\bot}(S)$ is $d$. 2. 2.

For any $Q_{1}$ and $Q_{2}\in\mathrm{Ann}^{\bot}(S)$, one has $Q_{1}\mathop{{}_{{}^{\sqcup\!\sqcup}}}Q_{2}\in\mathrm{Ann}^{\bot}(S)$. 3. 3.

For any $P\in{\mathbb{K}}\langle X\rangle$ and $Q_{1}\in\mathrm{Ann}^{\bot}(S)$, one has $P\triangleleft Q_{1}\in\mathrm{Ann}^{\bot}(S)$.

Definition 14

(fliessrealisation ) The formal power series $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ is differentially produced if there exist an integer $d$, a power series $f\in{\mathbb{K}}[\![\bar{q}_{1},\ldots,\bar{q}_{d}]\!]$, a homomorphism ${\cal A}$ from $X^{*}$ to the algebra of differential operators generated by

[TABLE]

such that, for any $w\in X^{*}$, one has $\langle S\mid w\rangle={\cal A}(w)\circ f_{|_{0}}$.

The pair $({\cal A},f)$ is called the differential representation of $S$ of dimension $d$.

Proposition 19

(reutenauerrealisation )* Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$. If $S$ is differentially produced then it satisfies the growth condition and its Lie rank is finite.*

Proof

Let $({\cal A},f)$ be a differential representation of $S$ of dimension $d$. Then, by the notations of Definition 10, we get

[TABLE]

We put

[TABLE]

Firstly, by Theorem 3.1, the generating series $\sigma f$ satisfies the growth condition.

Secondly, for any $\Pi\in{\cal L}ie_{{\mathbb{K}}}\langle X\rangle$ and for any $w\in X^{*}$, one has

[TABLE]

Since ${\cal A}(\Pi)$ is a derivation over ${\mathbb{K}}[\![\bar{q}_{1},\ldots,\bar{q}_{d}]\!]$:

[TABLE]

then we deduce that

[TABLE]

This means that $\sigma f\triangleright\Pi$ is a ${\mathbb{K}}$-linear combination of $\{T_{j}\}_{j=1,\ldots,d}$ and the dimension of the vector space $\mathrm{span}\{\sigma f\triangleright\Pi\mid\Pi\in{\cal L}ie_{{\mathbb{K}}}\langle X\rangle\}$ is less than or equal to $d$.

Fliess’ local realization theorem

Proposition 20

(reutenauerrealisation )* Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ with Lie rank $d$. Then there exists a basis ${S}_{1},\ldots,{S}_{d}\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ of $(\mathrm{Ann}^{\bot}(S),\mathop{{}_{{}^{\sqcup\!\sqcup}}})\cong({\mathbb{K}}[\![{S}_{1},\ldots,{S}_{d}]\!],\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ such that the ${S}_{i}$’s are proper and for any $R\in\mathrm{Ann}^{\bot}(S)$, one has*

[TABLE]

Proof

By Lemma 9, such a basis exists. More precisely, since the Lie rank of $S$ is $d$ then there exist $P_{1},\ldots,P_{d}\in{\cal L}ie_{{\mathbb{K}}}\langle X\rangle$ such that $S\triangleright P_{1},\ldots,S\triangleright P_{d}\in({\mathbb{K}}\langle\!\langle X\rangle\!\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ are ${\mathbb{K}}$-linearly independent. By duality, there exists ${S}_{1},\ldots,{S}_{d}\in({\mathbb{K}}\langle\!\langle X\rangle\!\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ such that

[TABLE]

Expanding this product, one obtains, via PBW theorem, the expected expression for the coefficients $\{r_{i_{1},\ldots,i_{d}}=\langle R\mid P_{1}^{i_{1}}\ldots P_{d}^{i_{d}}\rangle\}_{i_{1},\ldots,i_{d}\geq 0}$. Hence, $(\mathrm{Ann}^{\bot}(S),\mathop{{}_{{}^{\sqcup\!\sqcup}}})$ is generated by ${S}_{1},\ldots,{S}_{d}$.

With the notations of Proposition 20, one has

Corollary 6

If $S\in{\mathbb{K}}[{S}_{1},\ldots,{S}_{d}]$ then, for any $i=0,1$ and for any $j=1,\ldots,d$, one has $x_{i}\triangleleft S\in\mathrm{Ann}^{\bot}(S)={\mathbb{K}}[{S}_{1},\ldots,{S}_{d}]$. 2. 2.

The power series $S$ satisfies the growth condition if and only if, for any $i=1,\ldots,d$, ${S}_{i}$ also satisfies the growth condition.

Proof

Assume there exists $j\in[1,\ldots,d]$ such that ${S}_{j}$ does not satisfy the growth condition. Since $S\in\mathrm{Ann}^{\bot}(S)$ then using the decomposition of $S$ on ${S}_{1},\ldots,{S}_{d}$, one obtains a contradiction with the fact that $S$ satisfies the growth condition.

Conservely, using Proposition 16, we get the expected results.

Theorem 3.3

(fliessrealisation )* The formal power series $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ is differentially produced if and only if its Lie rank is finite and if it satifies the $\chi$-growth condition.*

Proof

By Proposition 19, one gets a direct proof. Conversely, since the Lie rank of $S$ equals $d$ then by Proposition 20, setting $\sigma f_{|_{0}}=S$ and, for $j=1,\ldots,d$, $\sigma\bar{q}_{i}={S}_{i}$,

We choose the observation $f$ as follows

[TABLE]

such that

[TABLE] 2. 2.

It follows that, for $i=0,1$ and for $j=1,\ldots,d$, the residual $x_{i}\triangleleft\sigma\bar{q}_{j}$ belongs to $\mathrm{Ann}^{\bot}(\sigma f_{|_{0}})$ (see also Lemma 9), 3. 3.

Since $\sigma f$ satisfies the $\chi$-growth condition then, the generating series $\sigma\bar{q}_{j}$ and $x_{i}\triangleleft\sigma\bar{q}_{j}$ (for $i=0,1$ and for $j=1,\ldots,d$) verify also the growth condition. We then take (see Lemma 8)

[TABLE]

by expressing $\sigma A_{j}^{i}$ on the basis $\{\sigma\bar{q}_{i}\}_{i=1,\ldots,d}$ of $\mathrm{Ann}^{\bot}(\sigma f_{|_{0}})$, 4. 4.

The homomorphism ${\cal A}$ is then determined as follows

[TABLE]

where, by Lemma 5, one has $A_{j}^{i}(\bar{q}_{1},\ldots,\bar{q}_{d})={\cal A}(x_{i})\circ\bar{q}_{j}$.

Thus, $({\cal A},f)$ provides a differential representation282828 In fliessrealisation ; reutenauerrealisation , the reader can find the discussion on the minimal differential representation. of dimension $d$ of $S$.

Moreover, one also has the following

Theorem 3.4

(fliessrealisation )* Let $S\in{\mathbb{K}}\langle\!\langle X\rangle\!\rangle$ be a differentially produced formal power series. Let $({\cal A},f)$ and $({\cal A}^{\prime},f^{\prime})$ be two differential representations of dimension $n$ of $S$. There exist a continuous and convergent automorphism $h$ of ${\mathbb{K}}$ such that*

[TABLE]

Since any rational power series satisfies the growth condition and its Lie rank is less than or equal to its Hankel rank which is finite fliessrealisation then

Corollary 7

Any rational power series and any polynomial over $X$ with coefficients in ${\mathbb{K}}$ are differentially produced.