In this paper, we give an explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers in terms of polylogarithm functions. addition, formulas will also be derived.

We present a new proof of result Knuth and Buckholtz concerning the period the number alternating congruences modulo an odd prime. The is based on properties special functions, specifically polylogarithm, Dirichlet eta beta Stirling numbers second kind.

Abstract As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions such functions and polynomials, were introduced by means the functions, some their properties investigated. The aim this paper to further study using three formulas coming from recently developed ‘ λ -umbral calculus’. In more detail, among other things, we rep...

We discuss inverse factorial series and their relation to Stirling numbers of the first kind. prove a special representation polylogarithm function in terms with such numbers. Using various identities for kind we construct number expansions functions where coefficients are These results used reprove asymptotic expansion some classical functions. also binomial formula involving factorials.

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that line is free from distortions, reduced to a similar one-dimensional inhomogeneous wave equation. An effective way solve this based on use of special functions – polylogarithms, which are complex power series with coefficients, converging unit circle. exact solution expressed integral...

We study sums S = S(d, n, k) = ∑ j≥1 [ d] jk( j )j! with d ∈ N = {1, 2, . . . } and n, k ∈ N0 = {0, 1, 2, . . . } and relate them to (finite) multiple zeta functions. As a byproduct of our results we obtain asymptotic expansions of ζ(d + 1) −H n as n tends to infinity. Furthermore, we relate sums S to Nielsen’s polylogarithm.

We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N=4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory ...

In this paper we investigate the representation of integrals involving product Legendre Chi function, polylogarithm function and log function. We will show that in many cases these take an explicit form Riemann zeta Dirichlet Eta lambda other special functions. Some examples illustrating theorems be detailed.

Fermi-Dirac and Bose-Einstein functions arise as quantum statistical distributions. The Riemann zeta function and its extension, the polylogarithm function, arise in the theory of numbers. Though it might not have been expected, these two sets of functions belong to a wider class of functions whose members have operator representations. In particular, we show that the Fermi-Dirac and Bose-Einst...