Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm

The prediction properties of Aitken's iterated Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant.Comment: 34 pages, LaTe

Shifted substitution in non-commutative multivariate power series with a view toward free probability

We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability

The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $(\mathbb Z_2^2)\wr\mathbb Z$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring $\mathbb Z_2[[t]]$ of formal power series over $\mathbb Z_2$.Comment: 18 pages, 2 figure

Topological Manin pairs and (n,s)(n,s)-type series

Lie subalgebras of $L = \mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^n\mathfrak{g}[x]$, complementary to the diagonal embedding $\Delta$ of $\mathfrak{g}[\![x]\!]$ and Lagrangian with respect to some particular form, are in bijection with formal classical $r$-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series $\mathfrak{g}[\![x]\!]$. In this work we consider arbitrary subspaces of $L$ complementary to $\Delta$ and associate them with so-called series of type $(n,s)$. We prove that Lagrangian subspaces are in bijection with skew-symmetric $(n,s)$-type series and topological quasi-Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$. Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type $(n,s)$, solving the generalized Yang-Baxter equation, correspond to subalgebras of $L$. We discuss their possible utility in the theory of integrable systems

Deformations of shuffles and quasi-shuffles

International audienceWe investigate deformations of the shuffle Hopf algebra structure Sh(A) which can be defined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffle algebra QSh(A), can be interpreted as natural transformations of the functor Sh, regarded as a functor from commutative nonunital algebras to coalgebras. We prove that the monoid of natural endomophisms of the functor Sh is isomorphic to the monoid of formal power series in one variable without constant term under composition, so that in particular, its natural automorphisms are in bijection with formal diffeomorphisms of the line. These transformations can be interpreted as elements of the Hopf algebra of word quasi-symmetric functions WQSym, and in turn define deformations of its structure. This leads to a new embedding of free quasi-symmetric functions into WQSym, whose relevance is illustrated by a simple and transparent proof of Goldberg's formula for the coefficients of the Hausdorff series

Déformations des algèbres de battages et quasi-battages

International audienceWe investigate deformations of the shuffle Hopf algebra structure Sh(A) which can be defined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffle algebra QSh(A), can be interpreted as natural transformations of the functor Sh, regarded as a functor from commutative nonunital algebras to coalgebras. We prove that the monoid of natural endomophisms of the functor Sh is isomorphic to the monoid of formal power series in one variable without constant term under composition, so that in particular, its natural automorphisms are in bijection with formal diffeomorphisms of the line. These transformations can be interpreted as elements of the Hopf algebra of word quasi-symmetric functions WQSym, and in turn define deformations of its structure. This leads to a new embedding of free quasi-symmetric functions into WQSym, whose relevance is illustrated by a simple and transparent proof of Goldberg's formula for the coefficients of the Hausdorff series.On s’intéresse aux déformations de la structure d’algèbre de Hopf des battages (ou shuffles) ${\rm Sh}(A)$ définie sur l’algèbre tensorielle sur une algèbre commutative $A$. Ces déformations, dont un cas remarquable est donné par l’algèbre de Hopf des quasi-shuffles ${\rm QSh}(A)$, s’interprètent comme transformations naturelles du foncteur ${\rm Sh}$ vu comme foncteur des algèbres commutatives non unitaires vers les coalgèbres. On montre en particulier que le monoïde des endomorphismes naturels du foncteur ${\rm Sh}$ est isomorphe au monoïde des séries formelles en une variable sans terme constant pour la loi de composition des séries. Les automorphismes naturels du foncteur sont donc en bijection avec les difféomorphismes formels de la droite.Ces transformations s’interprètent aussi comme des élements de l’algèbre de Hopf des surjections (ou, de façon équivalente des fonctions quasi-symétriques en mots) $\mathbf{WQSym}$, et en définissent à leur tour des déformations. Cette remarque conduit entre autres à un nouveau plongement des fonctions quasi-symétriques libres dans $\mathbf{WQSym}$ dont la pertinence est illustrée par une preuve simple de la formule de Goldberg pour les coefficients de la série de Hausdorff