q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

New heterocycles from 8‐hydroxyquinoline via dipolar 1,3‐cycloadditions: Synthesis & biological evaluation

Diarylnitrilimine and arylnitriloxide dipoles react with two 8‐hydroxyquinoline substrates to give respectively pyrazolinic and isoxazolinic derivatives. The structure of these new heterocycles was established on the basis of their spectroscopic data and by chemical methods. The inhibition activity of one of these heterocycles was evaluated in vitro against 8 pathogenic μ‐organisms.     

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.     

Synthesis of novel 5,7‐disubstituted 8‐hydroxyquinolines

Seven new 5,7‐disubstituted oxine derivatives have been synthesized via a Mannich reaction between a sec. amine (e.g. piperidine, pyrrolidine, morpholine, or dibenzylamine,) and 5‐cyano or 5‐azidomethyl‐8‐hydroxyquinoline, which were respectively obtained by nucleophilic displacement of 5‐chloromethyl‐8‐hydroxyquinoline by cyanide or azide anions. In all cases, a single product was isolated in medium to fair yield and characterized on the basis of ¹H and ¹³C‐NMR, MS and IR spectrometric data. The X‐ray structure of the product obtained from 5‐cyanomethyl‐8‐hydroxyquinoline and piperidine is also reported.     

New characterization of Appell polynomials

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli–Euler's one.     

Higher dimensional Dedekind sums in finite fields

We introduce Dedekind sums of a new type defined over finite fields. These are similar to the higher dimensional Dedekind sums of Zagier. The main result is the reciprocity law for them.     

Reciprocity formulae for generalized Dedekind‐Vasyunin‐cotangent sums

Recently, several works are done on the generalized Dedekind‐Vasyunin sum where and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum which is a new reciprocity law for the sum . Our result is a complement to recent results dealing with the sum studied by Bettin‐Conrey and then by Auli‐Bayad‐Beck. Accidentally, when a = 0, our reciprocity formula improves the known result in a previous study.     

Reciprocity formulae for multiple Dedekind–Rademacher sums

We introduce multiple Dedekind–Rademacher sums, in terms of values of Bernoulli functions, that generalize the classical Dedekind–Rademacher sums. The aim of this paper is to give and prove a reciprocity law for these sums. The main theorem presented in this paper contains all previous results in the literature about Dedekind–Rademacher sums.