Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

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.     

Bilateral generating functions and operational methods

Bilateral generating functions are those involving products of different types of polynomials. We show that operational methods offer a powerful tool to derive these families of generating functions. We study cases relevant to products of Hermite polynomials with Laguerre, Legendre and other polynomials. We also propose further extensions of the method which we develop here.     

N指标d维广义Wiener过程象集的m项代数和的几个性质

设珮犠（狋）：犚犖＋ →犚犱是犖指标犱维广义Ｗｉｅｎｅｒ过程，对任意紧集犈１，…，犈犿犚犖＞ ，该文研究了犿项代数和珮犠（犈１）…珮犠（犈犿）的Ｈａｕｓｄｏｒｆｆ维数，Ｐａｃｋｉｎｇ维数和正的Ｌｅｂｅｓｇｕｅ测度及内点的存在性． 其结果包含并推广了布朗单的结果．     

Algebraic methods in quantum mechanics: from molecules to polymers

We present a brief review of algebraic techniques developed and applied in molecular spectroscopy in the last five years. We also outline perspectives for new applications of the Lie algebraic method in the first decade of the new century. Received 21 November 2001     

A new approach to obtain the non-Condon factors in closed form for two one-dimensional harmonic oscillators

A simple algebraic approach to calculate general Franck-Condon overlaps is extended to evaluate non-Condon factors for two one-dimensional harmonic oscillators. The method is based on the use of eigenstates of the harmonic oscillator annihilation operator which allows to obtain in terms of a multi-dimensional Hermite polynomial the overlap of harmonic oscillator functions associated with different Born-Oppenheimer potentials. The presented approach is self-contained, only basic concepts of quantum mechanics associated with the harmonic oscillator system are needed. The obtained expression for the Franck-Condon overlaps is similar to the Ansbacher’s formula and equivalent to the one calculated by Malkin and Man’ko. However our final expression has the advantages that only real numbers are involved and it is straightforward to get the limit case of equal frequencies. Concerning the non-Condon factors two approaches leading to different formulas are considered, both of which reduce to triple sums of products of three Hermite polynomials.     

ON THE RATE OF WEIGHTED Lp CONVERGENCE BY THE GRUENWALD INTERPOLATORY OPERATORS

This paper gives the weighted Lp convergence rate estimations of the Gruenwald interpolatory polynomials based on the zeros of Chebyshev polymomials of the first kind,and proves that the order of the estimations is optimal for p≥1.