Mesure de Mahler et calcul de constantes universelles pour les polynomes deN variables

Sans résumé

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Ideaux de polynomes et ideaux de valeurs

Let B be the ring of integral valued polynomials over a noetherian domain A. We study in which case finitely generated ideals of B are uniquely determined by their ideals of values at each element of A. We give necessary and sufficient conditions which are verified for example when A is any ring of integers of an algebraic number field, such that each quotient ring Am with respect to a maximal ideal m is analytically irreducible.     

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field or let M be the space k ⁿ and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H _·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C _·(A,A),b) and (^·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

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Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire     

Sur une classe de polynomes généralisant les polynomes de legendre

Sans résumé Présenté par P. Turán     

Suites de toeplitz,p-pliage,suites automatiques et polynomes

Sans résumé     

Some new asymptotic properties for the zeros of Jacobi,Laguerre, and Hermite polynomials

For the generalized Jacobi, Laguerre, and Hermite polynomials $P_n^{\left( {\alpha _n ,\beta _n } \right)} \left( x \right),L_n^{\left( {\alpha _n } \right)} \left( x \right),H_n^{\left( {\gamma _n } \right)} \left( x \right)$ , the limit distributions of the zeros are found, when the sequences α _n or β _n tend to infinity with a larger order thann. The derivation uses special properties of the sequences in the corresponding recurrence formulas. The results are used to give second-order approximations for the largest and smallest zero which improve (and generalize) the limit statements in a paper by Moak, Saff, and Varga [11].     

Formes de Jacobi et formules de distribution

The main theorem proved in this paper consists of a multiplicative distribution formula for the Jacobi forms in two variables associated to Klein forms. This gives stronger versions of distribution formulae appearing in the literature. Indeed, as a first consequence of the main theorem, we deduce an optional proof of the distribution formula true for any elliptic function first found by Kubert and as a second consequence, we prove an ameliorated distribution formula for a certain zeta function previously treated by Coates, Kubert and Robert. Moreover, our main theorem provides the exact root of unity appearing in the distribution formula of Jarvis and Wildeshaus, a fact which could be useful in the K-theory of elliptic curves or more precisely, in the investigation of the elliptic analogue of Zagier's conjecture linking regulators and polylogarithms.     

Polynome de Hilbert-Samuel dans les algebres enveloppantes et les algebres de groupe

An analogue of the Hilbert-Samuel polynomial is considered with respect to the augmentation ideal of the enveloping algebra of a finite dimensional nilpotent Lie algebra and the group ring of a finitely generated, torsion-free nilpotent group. Then, the Hilbert series of finitely generated modules are rational.      

Sur les suites de Laplace et sur les congruencesW

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The Combinatorics of Laguerre,Charlier, and Hermite Polynomials

This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so-called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three-term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.