The Double Laplace Transform Expressed in terms of the Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

A Note on Some Reduction Formulas for the Incomplete Beta Function and the Lerch Transcendent

We derive new reduction formulas for the incomplete beta function Bν,0,z and the Lerch transcendent Φz,1,ν in terms of elementary functions when ν is rational and z is complex. As an application, we calculate some new integrals. Additionally, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.

Around the Lipschitz Summation Formula

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

Chains of Truncated Beta Distributions and Benford’s Law

It was proved by Jang et al. that various chains of one-parameter distributions converge to Benford’s law. We study chains of truncated distributions and propose another approach, using a recent convergence result of the Lerch transcendent function, to proving that they converge to Benford’s law for initial Beta distributions with parameters α and 1.

Evaluation of integrals with hypergeometric and logarithmic functions

We provide an explicit analytical representation for a number of logarithmic integrals in terms of the Lerch transcendent function and other special functions. The integrals in question will be associated with both alternating harmonic numbers and harmonic numbers with positive terms. A few examples of integrals will be given an identity in terms of some special functions including the Riemann zeta function. In general none of these integrals can be solved by any currently available mathematical package.

A diagonal recurrence relation for the Stirling numbers of the first kind

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Polylogarithmic zeta functions and their p-adic analogues

We consider a broad family of zeta functions which includes the classical zeta functions of Riemann and Hurwitz, the beta and eta functions of Dirichlet, and the Lerch transcendent, as well as the Arakawa–Kaneko zeta functions and the recently introduced alternating Arakawa–Kaneko zeta functions. We construct their [Formula: see text]-adic analogues and indicate the many strong connections between the complex and [Formula: see text]-adic versions. As applications, we focus on the alternating case and show how certain families of alternating odd harmonic number series can be expressed in terms of Riemann zeta and Dirichlet beta values.

Some Properties of a Solution to a Family of Inhomogeneous Linear Ordinary Differential Equations

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.