On representations and differences of Stieltjes coefficients, and other relations

Certain logarithmic integrals, including solution of Monthly problem #tbd, zeta values, and expressions for the Stieltjes constants

We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values …

具有多项式源的广义Stieltjes系统

A generalized Stieltjes system with polynomial source

研究具有多项式源的广义Stieltjes系统，证明解数不超过二项式系数，并给出达到上界的条件，同时描述大线性项系数时的解结构。

Hypergeometric summation representations of the Stieltjes constants

The Stieltjes constants $γ_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the p…

Stieltjes-Bethe equations in higher genus and branched coverings with even ramifications

We describe projective structures on a Riemann surface corresponding to monodromy groups which have trivial $SL(2)$ monodromies around singularities and trivial $PSL(2)$ monodromies along homologically non-trivial loops on a Riemann surface. We propose a natural higher genus analog of Stieltjes-Bethe equations. Links with branched projective structures and with Hurwitz spaces with ramifications of even order are established. We find a higher genus analog of the genus zero Yang-Yang function (the function generating accessory parameters) and describe its similarity and difference with Bergman t…

On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions

We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta func…

变形Laguerre-Hahn正交多项式的可积微分系统

Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials

本文研究带变形参数的Laguerre-Hahn正交多项式，通过推导Lax对得到递推系数的非线性微分方程，并证明一个非半经典特例由Painlevé VI型方程支配，其参数与Magnus解相同但边界条件不同。

Exact treatment of linear difference equations with noncommutative coefficients

The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.

Exact treatment of operator difference equations with nonconstant noncommutative coefficients

In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative expression of a generic term of the unknown sequence of vectors of the Hilbert space. Some non-trivial applications are reported with the aim of showing the usefulness and the broad applicability of our result.

General solution of a second order non-homogenous linear difference equation with noncommutative coefficients

The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying some applications belonging to different mathematical contexts.

具有奇异扰动 Laguerre 型权函数的正交多项式递推系数的微分方程与差分方程

Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight

本文针对奇异扰动 Laguerre 型权函数下的首一正交多项式，采用阶梯算子方法建立递推系数的非线性二阶差分方程及微分-差分方程，并据此得到其大 n 渐近展开。

Spectral statistics for the difference of two Wishart matrices

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order …

On the difference equations with periodic coefficients

In this paper, we study entire solutions of the difference equation $ψ(z+h)=M(z)ψ(z)$, $z\in{\mathbb C}$, $ψ(z)\in {\mathbb C}^2$. In this equation, $h$ is a fixed positive parameter and $M: {\mathbb C}\to SL(2,{\mathbb C})$ is a given matrix function. We assume that $M(z)$ is a $2π$-periodic trigonometric polynomial. We construct the minimal entire solutions, i.e. entire solutions with the minimal possible growth simultaneously as for im$z\to+\infty$ so for im$z\to-\infty$. We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the s…

Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem

A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of such solutions leads to a notion of local monodromies of difference equations. It is shown that in the continuous limit they converge to the monodromy matrices of differential equations. New type of isomonodromic deformations of difference equations with elliptic coefficients changing the periods of elliptic curves is constructed.

对于具有变量系数的抛物型和Schrödinger型方程的紧致差分格式

Compact difference scheme for parabolic and Schrödinger-type equations with variable coefficients

本文提出了一种新的紧致格式，用于求解具有变量时间不变系数的二阶偏微分方程（抛物型和Schrödinger型）。该格式具有更高的阶数和更小的误差，相较于经典隐式格式。考虑了狄利克雷和Neumann边界问题。相对有限差分算子几乎为自共轭的。

One-point commuting difference operators of rank one

We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of hyperelliptic spectral curve when the marked point coincides with the branch point. We construct examples of operators with polynomial and trigonometric coefficients. Moreover, difference operators with polynomial coefficients can be embedded in the differential ones with polynomial coefficients. This construction provides a new way of constructing commutative su…