Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying these formulas, we can prove that multiple zeta star values whose indices are the sequences $(\bar 1,\{1\}_m,\bar 1)$ and $(2,\{1\}_m,\bar 1)$ can be expressed polynomially in terms of zeta values, polylogarithms and $\ln2$. Finally, ...

Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their values.

We give systematic method to evaluate a large class of one-dimensional integral relating to multiple zeta values (MZV) and colored MZV. We also apply the technique of iterated integrals and regularization to elucidate the nature of some infinite series involving binomial coefficients. This technique can be applied to many Ap\'ery-type infinite sums.

Using properties of the Riemann zeta-function we propose two new large classes of evaluated series. Incidentally the first class represents integrals as generalized average on very nonuniform sequences. The second class contains inter alia a lot of new series with the Jacoby theta-functions and rationals of the exponential function. Moreover we propose many functions that can replace the Riemann zeta-function in similar constructions. Two examples: 1) if $f(x)$ has period 1...

In this paper we investigate a class of integrals that were encountered in the course of a work on statistical plasma physics, in the so-called Sommerfeld temperature-expansion of the electronic entropy. We show that such integrals, involving some parameters, can be fully described in closed form represented by special functions.

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent, $\Phi(e^m,-k,b)$), that coincide with their Abel-Plana expressions. A slight variation of the approach leads to different formulae. We also present the relations between each of these functions and their partial sums. It allows one to figure, for example, t...

We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p /(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q \in\N$). The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either $\zeta^{(p)}(\pm q)$ (first two sums), 0 (third sum) or yield novel mathematical constants (fo...

This paper describes generalized polylogarithms, multiple polylogarithms, and multiple zeta values, along with their implementation in Maple 2018. This set of related functions is of interest in high energy physics as well as in number theory. Algorithms for the analytical manipulation and numerical evaluation of these functions are described, along with the way these features are implemented in Maple.

Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re(s)>1-r$, $p\geq 1$, and $x+1>0$. The zeta function $Z^{\bf\alpha}_p(s;x)$ is defined by $$ Z^{\bf\alpha}_p(s;x) =\frac{1}{\Gamma(s)}\int^\infty_0 \frac{e^{-xt}} {e^t-1}\,Li_{\bf{\alpha}}\left(\frac{1-e^{-t}}p\right) t^{s-1}\,dt, $$ where ${\bf\alpha}=(\alpha_1,\ldots,\alpha_r)$ is a $r$-tuple positive integers, and $Li_{\bf{\alpha}}(z)$ is the one-variable multiple polylogarithms. Since $Z^{\bf\alpha}_1(s;0)=\x...

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then expressed 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.