Khristo N. Boyadzhiev
We introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the logGamma function. This technique is based on Hankel integral representations of the Hurwitz zeta, the Lerch Zeta, the Digamma and logGamma functions.
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The main aim of this paper is to give a new generalization of Hurwitz-Lerch Zeta function of two variables.Also, we investigate several interesting properties such as integral representations, summation formula and a connection with generalized hypergeometric function. To strengthen the main results we also consider many important special cases.
The main objective of this paper is to introduce a new extension of Hurwitz-Lerch Zeta function in terms of extended beta function. We then investigate its important properties such as integral representations, differential formulas, Mellin transform and certain generating relations. Also, some important special cases of the main results are pointed out.
Rational approximation to the Hurwitz-Lerch zeta function
本文基于作者1996年提出的“余项Padé逼近”方法的推广，构造了Lerch函数的有理逼近新序列，并应用于Stieltjes常数的逼近。
We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers …
In this paper, we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion.
We find a representation for the Maclaurin coefficients of the Hurwitz zeta-function in terms of semi-convergent series involving the Bernoulli polynomials and the Stirling numbers of the first kind. In particular, this gives a representation for the coefficients of the Riemann zeta function. Our main instrument is a certain series transformation formula. A similar result is proved also for the Maclaurin coefficients of the Lerch zeta function.
The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} ζ_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $ζ_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.
On gamma functions with respect to the alternating Hurwitz zeta functions
本文进一步研究由交替Hurwitz zeta函数定义的gamma函数$\widetildeΓ(x)$，获得了与经典gamma函数类似的多种性质，并证明了一个Lerch型公式，表明$ζ_{E}(z,x)$的导数可由$\widetildeΓ(x)$表示。
Explicit Formulas For Generalized Polylogarithmic Integrals, Euler Sums, And BBP-Type Series
本文研究含五参数的多对数积分闭式表达式，用Lerch超越函数统一表示，并讨论其与欧拉和及BBP型级数的联系。
In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $π$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.
On the Coefficients of Hurwitz-Type Matrix Polynomials
该研究推导了赫维茨型矩阵多项式系数的显式公式，引入块赫维茨矩阵并建立相关行列式恒等式，还证明其系数矩阵行列式正定性在次数≤3时成立、次数为4时不成立，解决了相关猜想。
Complex zeros of Bessel function derivatives and associated orthogonal polynomials
本文构造以涉及贝塞尔导数 $J_ν'$ 零点的 Rayleigh 型和为矩的正交多项式，研究其递推与正交性并给出 Hankel 行列式公式，进而完善 $J_ν'$ 复零点数随 ν 变化的 Hurwitz 型定理。
Computing Stieltjes constants using complex integration
本文提出一种高效算法，通过复积分计算Stieltjes常数，实现任意精度计算并保证误差界，首次以低复杂度计算高阶项。
A Resurgent Analytic Framework for Indicial Umbral Calculus via Mellin-Barnes and Borel-Laplace Theories
通过Mellin-Barnes积分、Borel-Laplace求和与复兴分析，为指示性阴影微积分建立严格分析基础，将阴影算子提升为连续线性泛函，揭示形式代数恒等式为全局解析对应的局部展开。
The Parabolic Mellin Transform: Gamma and Zeta Integral Representations
本文介绍了抛物线梅林变换（PMT），并通过其对高斯核的处理，得到经典汉克尔表示的抛物线轮廓形式，并利用该变换推导了Hurwitz泽塔函数、黎曼泽塔函数和狄利克雷eta函数的积分表示，同时提供了Gamma型和泽塔型特殊函数的统一变换字典，并将黎曼假设和林德洛夫假设重新表述为抛物线轮廓积分零点和增长的等价形式。
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