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In this paper, we study the almost Bézout property in different commutative ring extensions, namely, in bi-amalgamated algebras and pairs of rings. In Section 2, we deal with almost Bézout domains issued from bi-amalgamations. Our results capitalize well known results on amalgamations and pullbacks as well as generate new original class of rings satisfying this property. Section 3 investigates pairs of rings where all intermediate rings are almost Bézout domains. As an application of our results, we characterize pairs of rings $(R,T)$, where $R$ arises from a $(T,M,D)$ construction to be an almost Bézout domain.

In this paper, we discuss new results related to the generalized discrete q-Hermite II polynomials h ˜ n , α ( x ; q ) , introduced by Mezlini et al. in 2014. Our aim is to give a continuous orthogonality relation, a q-integral representation and an evaluation at unity of the Poisson kernel, for these polynomials. Furthermore, we introduce q-Schrodinger operators and we give the raising and lowering operator algebra corresponding to these polynomials. Our results generate a new explicit realization of the quantum algebra su q ( 1 , 1 ) , using the generators associated with a q-deformed generalized para-Bose oscillator.

Resume Soit Γ ⊂ PSL ( 2 , R ) un sous-groupe discret de covolume fini. On suppose que la courbe modulaire H / Γ se « plonge » dans une surface modulaire de Hilbert H 2 / Γ K , ou Γ K est le groupe modulaire de Hilbert associe a un corps quadratique reel K . On definit une suite de restrictions ( ρ n ) n ∈ N des formes modulaires de Hilbert pour Γ K aux formes modulaires pour Γ . Notons M k 1 , k 2 ( Γ K ) l'espace des formes modulaires de Hilbert de poids ( k 1 , k 2 ) pour Γ K . On demontre que ∑ n ∈ N ∑ k 1 , k 2 ∈ N ρ n ( M k 1 , k 2 ( Γ K ) ) est un sous-espace stable par les crochets de Rankin–Cohen de l'espace ⊕ k ∈ N M k ( Γ ) des formes modulaires pour Γ . Pour citer cet article : N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Resume On decrit la structure additive de l'anneau gradue M ˜ ∗ ( Γ ) des formes quasi-modulaires sur un groupe discret et co-compact Γ ⊂ PSL ( 2 , R ) . On montre que cet anneau n'est pas de type fini. On calcule le nombre de generateurs nouveaux en chaque poids k (pair). Le nombre en question est fixe pour k assez grand, et est egal a dim C I / ( I ∩ I ˜ 2 ) ou I et I ˜ designent les ideaux des formes modulaires, respectivement quasi-modulaires, sur Γ en poids strictement positif. Pour citer cet article : N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form $\sum_{n\leq x}\Lambda (n)\exp(2i\pi f(n+c_n)+\beta n)$ where $f$ a digital function, $\mathbf{c}=(c_n)$ is an almost-periodic sequence in $ \mathbb{Z}$ and $\beta $ is a real parameter, which extend the works of Mauduit-Rivat \cite{mr1} and Martin-Mauduit-Rivat \cite{mmr} to the case of the shifted prime numbers satisfying a digital constraint.

We describe the additive structure of the graded ring $\widetilde{M}_*$ of quasimodular forms over any discrete and cocompact group $\Gamma \subset \rm{PSL}(2, \RM).$ We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight $k$. This number is constant for $k$ sufficiently large and equals $\dim_{\CM}(I / I \cap \widetilde{I}^2),$ where $I$ and $\widetilde{I}$ are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that $\widetilde{M}_*$ is contained in some finitely generated ring $\widetilde{R}_*$ of meromorphic quasimodular forms with $\dim \widetilde{R}_k = O(k^2),$ i.e. the same order of growth as $\widetilde{M}_*.$, Comment: 22 pages, 1 figure