Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{\left(\alpha \right)}\left(x\right)$ vanishes if and only if $n\ge \left|\alpha \right|=?\alpha$. Therefore we assume that $\alpha =?n?s?1$ where $s$ is a non-negative integer. Let $g\left(x\right)={(?1)}^n{L}_n^{(?n?s?1)}\left(x\right)=\sum _{j=0}^n{a}_j\frac{x^j}{j!}$ and more general polynomial, let $G\left(x\right)=\sum _{j=0}^n{a}_j{b}_j\frac{x^j}{j!}$ where $b_j$ with $0\le j\le n$ are integers such that $\left|b_0\right|=\left|b_n\right|=1$. Schur was the first to prove the irreducibility of $g\left(x\right)$ for $s=0$. It has been proved that $g\left(x\right)$ is irreducible for $0\le s\le 60$. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either $G\left(x\right)$ is irreducible or $G\left(x\right)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s>1.9k$ whenever $G\left(x\right)$ has a factor of degree $k\ge 2$ and $(n,k,s)\ne (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.     

Theory of chemical bonds in metalloenzymes XXII: a concerted bond-switching mechanism for the oxygen–oxygen bond formation coupled with one electron transfer for water oxidation in the oxygen-evolving complex of photosystem II

ABSTRACT

QM(UB3LYP)/MM(AMBER) calculations were performed for the locations of the transition structure (TS) of the oxygen–oxygen (O–O) bond formation in the S₄ state of the oxygen-evolving complex (OEC) of photosystem II (PSII). The natural orbital (NO) analysis of the broken-symmetry (BS) solutions was also performed to elucidate the nature of the chemical bonds at TS on the basis of several chemical indices defined by the occupation numbers of NO. The computational results revealed a concerted bond switching (CBS) mechanism for the oxygen–oxygen bond formation coupled with the one-electron transfer (OET) for water oxidation in OEC of PSII. The orbital interaction between the σ-HOMO of the Mn(IV)₄–O₍₅₎ bond and the π*-LUMO of the Mn(V)₁=O₍₆₎ bond plays an important role for the concerted O–O bond formation for water oxidation in the CaMn₄O₆ cluster of OEC of PSII. One electron transfer (OET) from the π-HOMO of the Mn(V)₁=O₍₆₎ bond to the σ*-LUMO of the Mn(IV)₄–O₍₅₎ bond occurs for the formation of electron transfer diradical, where the generated anion radical [Mn(IV)₄–O₍₅₎]^-? part is relaxed to the ?Mn(III)₄?…?O₍₅₎^- structure and the cation radical [O₍₆₎=Mn(V)₁]⁺ ? part is relaxed to the ⁺O₍₆₎–Mn(IV)₁? structure because of the charge-spin separation for the electron-and hole-doped Mn–oxo bonds. Therefore, the local spins are responsible for the one-electron reductions of Mn(IV)₄->Mn(III)₄ and Mn(V)₁->Mn(IV)₁. On the other hand, the O₍₅₎^- and O₍₆₎⁺ sites generated undergo the O–O bond formation in the CaMn₄O₆ cluster. The Ca(II) ion in the cubane- skeleton of the CaMn₄O₆ cluster assists the above orbital interactions by the lowering of the orbital energy levels of π*-LUMO of Mn(V)₁=O₍₆₎ and σ*-LUMO of Mn(IV)₄–O₍₅₎, indicating an important role of its Lewis acidity. Present CBS mechanism for the O–O bond formation coupled with one electron reductions of the high-valent Mn ions is different from the conventional radical coupling (RC) and acid-base (AB) mechanisms for water oxidation in artificial and native photosynthesis systems. The proton-coupled electron transfer (PC-OET) mechanism for the O–O bond formation is also touched in relation to the CBS-OET mechanism.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on C~n Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Given a principal value convolution on the Heisenberg group Hn = Cn×R, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on Cn. We also calculate the Dirichlet kernel for the Laguerre expansion on the group Hn.     

A new summation identity for the Srivastava-Singhal polynomials

In his recent investigations involving differential operators for some generalizations of the classical Laguerre polynomials, H. Bavinck [J. Phys. A Math. Gen. 29 (1996) L277-L279] encountered and proved a certain summation identity for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation identity for the Srivastava-Singhal polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. It is also indicated how the general summation identity can be applied to derive the corresponding result for one class of the Konhauser biorthogonal polynomials.     

Some families of hypergeometric polynomials and associated integral representations

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single-, double-, and triple-integral representations. Some known or new consequences of the general results presented here, involving such classical orthogonal polynomials as the Jacobi, Laguerre, Hermite, and Bessel polynomials, and various other relatively less familiar hypergeometric polynomials, are also considered. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Multilayered Cu–Ti deposition on silicon substrates for chemiresistor applications

Abstract

On the perspective to develop CuO–TiO₂ MOS, multilayered Cu and Ti thin layers were alternatively deposited on silicon wafers using 25?keV Ar?+?ion beam sputtering and, subsequently, oxidized by thermal annealing in air at 400?°C for 24?h. The deposited films have variable ratios of the Cu and Ti % at. One of the main goal is to obtain such multilayers avoiding the presence of Cu–Ti–O compounds. The samples were characterized in terms of composition (by RBS and SIMS analyses) and morphology (by AFM and SEM investigations). In particular, SIMS maps allows to observe the spatial distribution and thickness of each phase of the Cu/Ti multilayers, and further to observe Cu diffusion and mixing with Ti, as well as phase separation of CuO and TiO₂ in the samples. The reasons of this effect represent an open issue that has to investigated, in order to improve the MOS fabrication.     

Reaction des lithiens avec les organophosphores. Reaction d'addition du t-butyllithium sur le noyau benzenique des sels de phenylphosphoniums

A new reaction of organolithium compounds with phosphonium salts is described; reaction of t-butyllithium with dibenzylaminophosphonium or tetraphenylphosphonium bromides takes place through addition of the t-butyl group on a benzene ring at the position para to the phosphorus atom, the ylid formed reacts in a characteristic Wittig reaction with p-tolylaldehyde.     

Two-dimensional catena–polymer [[[bis (µ-aqua)(bis (µ-cpiap-K2O1:O2)sodium(I))] (µ-chlorido) bis(µ-cpiap-K5N1:O1:O2:O3,O3′)dicopper(II)]bis(µ-aqua)(aqua)bis(cpiap-K2O1,O2)copper(II)] hexa hydrate

The new title two-dimensional hetero-tetra nuclear Cu₃–Na coordination polymer {[NaCu₃Cl(cpiap)₂(H₂O)₃]_n·6nH₂O} (1) consists of crystallographically two-independent copper(II) centers, each bridged by a sodium cation through carboxylate-oxygen of the deprotonated H₃cpiap ligand (H₃cpiap = 2-(carboxyphenyl)iminoaceticpropanoic acid) to Cu^II (2) and Cu^II (2⁻) cations, and through water molecules to Cu^II (1) cation. Cu^II (2) and Cu^II (1) cations are bridged by carboxylate-oxygen atoms of the ligand in a syn-anti mode which, alternate regularly within the chain being bridged by a tetra coordinated sodium cation. Each Cu^II (2) and Cu^II (2⁻) cation in (1) is in an octahedral environment formed by four carboxylate-oxygens from two cpiap³⁻ ligands, one nitrogen atom and a bridging chloride atom. Cu^II (1) cation is in a square pyramidal environment formed by three water molecules and two carboxylate-oxygens from two cpiap³⁻ ligands. The ligand acts simultaneously as monodentate and tridentate toward Cu^II (1) and Cu^II (2) cations respectively. The lattice water molecules involved in OH···O hydrogen bonding are situated in the void spaces between layers. The zigzag chains, which run along the b-axes further construct three-dimensional metal-organic framework via hydrogen bonding and weak face-to-face π-π interactions. Weak CH···O interactions are also present.