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.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

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.     

变系数非线性方程的Jacobi椭圆函数展开解

把Jacobi椭圆函数展开法扩展并应用到求解变系数的非线性演化方程,比较方便地得到新的解析解 关键词： Jacobi椭圆函数 变系数非线性方程 类椭圆余弦波解 类孤子解     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Singularity conjecture and constructions of Jacobi forms

This paper verifies the singularity conjecture for Jacobi forms with higher degree in some typical cases, and gives constructions for the Jacobi cusp forms whose Fourier coefficients can be expressed by some kind of Rankin-typeL-series.