István Mező
Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for the $Ω=W(1)$ constant as well.
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We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion formulas are proved in a Banach space of functions whose Mellin's transforms are integrable over the vertical line Re s > 1. Examples of new transformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited. Particular cases include familiar Lambert and Mobius transformations. Finally a class of equivalences of the Salem type to the Riemann hypothes…
On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences
本研究利用Mellin变换处理两类特殊形式的级数，推导其积分表示并进行变换，得到相关积分结果、极限与超幂级数表示，还推广了Riemann关于Zeta、Gamma与Jacobi Theta函数的经典关系式。
We study some series expansions for the Lambert $W$ function. We show that known asymptotic series converge in both real and complex domains. We establish the precise domains of convergence and other properties of the series, including asymptotic expressions for the expansion coefficients. We introduce an new invariant transformation of the series. The transformation contains a parameter whose effect on the domain and rate of convergence is studied theoretically and numerically. We also give alternate representations of the expansion coefficients, which imply a number of combinatorial identiti…
Employing the Lagrange inverting series, a solution of the transcendental equation $(x-a)(x-b)=le^{x}$, that can be considered a quadratic generalization of the equation defining Lambert $W$ function, has been found in terms of Bessel orthogonal polynomials. Once again a transcendental equation can be formally solved by means of classic orthogonal polynomials, suggesting a link between Rodrigues formulas and the terms of Lagrange series. A novel representation for Bessel polynomials has been found, by means of differential identity : $\left(x^{2}D\right)^{n}=x^{n+1}D^{n}x^{n-1}$
On a generalised Lambert $W$ branch transition function arising from $p,q$-binomial coefficients
针对高维 Lenz-Ising 模型磁化分布建模需求，本文研究由 $p,q$-二项式系数引出的广义 Lambert $W$ 分支转移函数 $\omega$，并给出其分支显式公式及多种解析性质。
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $ζ(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular tran…
In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{F}_{k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{extended higher He…
An explicit transformation for the series $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1},$ Re$(y)>0$, which takes $y$ to $1/y$, is obtained for the first time. This series transforms into a series containing $ψ_1(z)$, the derivative of Deninger's function $R(z)$. In the course of obtaining the transformation, new important properties of $ψ_1(z)$ are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$. Our transformation readily gives the complete asymptotic expansion of $\sum\limits_{n=1}^{…
Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions
本文研究 Berndt 型积分，将其置于 Barnes-zeta 函数框架下，建立其与 Jacobi 椭圆函数及矩多项式的联系，拓展并统一了相关特殊函数理论。
Evaluation of beta integrals of Ramanujan type and integral representations for bilateral hypergeometric series
本文评估Ramanujan型伽马函数乘积积分，表示为双边超几何级数，并在可求和时化为beta积分，同时给出双边超几何级数的积分表示。
This paper establishes a real integral representation of the reciprocal $Γ$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.
We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $χ_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta function $ζ(2 n+1)$ and the Dirichlet beta function $β(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations.
An integral representation is provided for the parabolic cylinder function product $D_μ(x)D_μ(-y)$ where $Re\,μ<0$ and $x>y$ are unrelated. A few simple consequences are given in the form of hyperbolic integrals and a sum rule.
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting …
We give an explicit integral formula for the Dunkl kernel associated to root system of type $A_2$ and parameter $k>0$, by exploiting recent result in [1].
The aim of this paper is two fold. We derive an integral representation for the generalized 2D Zernike polynomials which are of independent interest and give the explicit expression of the action of the Cauchy transform on them.
We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence.
Weighted norm estimates and representation formulas are proved for non-homogeneous singular integrals with no regularity condition on the kernel and only an L log L integrability condition. The representation formulas involve averages over a star-shaped set naturally associated with the kernel. The proof of the norm estimates is based on the representation formulas, some new variations of the Hardy-Littlewood maximal function, and weighted Littlewood-Paley theory.
Generalization of the integral representation of the gamma function has been obtained, which shows that the Hankel contour assumes rotation in the complex plane. The range of admissible values for the contour rotation angle is set. Using this integral representation, generalization of the integral representation of the Mittag-Leffler function has been obtained that expresses the value of this function in terms of the contour integral.
Integral representations for a generalized Hermite linear functional
该研究针对广义Hermite线性泛函，推导了其在实轴和复平面上的新积分表示，并将结果应用于得到Euler Gamma函数的新积分表示，丰富了相关特殊函数的积分表达形式。
Integral representation and functional inequalities involving generalized polylogarithm
本文用Dirichlet级数与Laplace表示及Hadamard卷积两种方法，得到广义多对数函数的单、双积分表示，并证明其完全单调性、Turan不等式、凸性与界，另给出Lerch超越函数积分表示及Hadamard卷积的替代证明。
The integral representation of the two-parameter Mittag-Leffler function $E_{ρ,μ}(z)$ is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function $E_{ρ,μ}(z)$ has two forms: the representation ``A'' and ``B''. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and…
We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this representation, we obtain new functional identities for the Riemann zeta function.
In this paper, we give a two dimensional analogue of the Euler-MacLaurin summation formula. By using this formula, we obtain an integral representation of Weil's elliptic functions which was introduced in the book "Elliptic functions according to Eisenstein and Kronecker".
We consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.
The paper presents an integral representation of the two-parameter Mittag-Leffler function $E_{ρ,μ}(z)$ and singular points of this representation have been studied. It has been found that there are two singular points for this integral representation: $ζ=1$ and $ζ=0$. The point $ζ=1$ is a pole of the first order and the point $ζ=0$, depending on the values of parameters $ρ,μ$ is either a pole or a branch point, or a regular point. The subsequent study showed that at some values of parameters $ρ,μ$ with the help of the residue theory one can calculate the integral included in the studied integ…
We prove that the functions Phi(x)=[Gamma(x+1)]^{1/x}(1+1/x)^x/x and log Phi(x) are Stieltjes transforms.
We show how the unformizable representation for Abelian integrals of nontrivial genera arise. The technique makes use of the famous Chudnovsky's Fuchsian linear differential equations and their relation to the sixth Painleve transcendent.
A Fourier-type integral representation for Bessel's function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson's integral representation for the Bessel function along with a trigonometric integral representation of Gegenbauer's polynomials. This representation lets us express various functions related to the incomplete gamma function in series of Bessel's functions. Neumann series of Bessel functions are also considered and a new closed-form integral representation for this class of series is given. The density function of this representation is simpl…
We give integral representations for multiple Hermite and multiple Hermite polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.
In the paper, the authors find some integral representations and discover some properties of Lah numbers.
We derive a novel integral representations of Jacobi polynomials in terms of the Gauss hypergeometric function. Such representation is then used to give the explicit integral representation for the Heat kernel on the quantized Riemann sphere.
In the paper, the author establishes an integral representation and properties of Bernoulli numbers of the second kind and reveals that the generating function of Bernoulli numbers of the second kind is a Bernstein function on $(0,\infty)$.
In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind.
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.
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