Integrals of polylogarithms and infinite series involving generalized harmonic numbers

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain fin...

Identities Involving Generalized Harmonic Numbers and Other Special Combinatorial Sequences

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Riordan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of s...

Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers

A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of special numbers. Starting with harmonic sums and polylogarithms we discuss recent extensions of these quantities as cyclotomic, generalized (cyclotomic), and bin...

Generalized multiple Dirichlet series and generalized multiple polylogarithms

where t is a complex variable. By Assumption I, we see that the series (1.3) is convergent when <t < 0. We further assume the following: (Assumption II ) ψ(s) can be continued analytically to the whole complex plane C, and holomorphic for all s ∈ C. In any fixed strip σ1 ≤ σ ≤ σ2, ψ(s; u) is uniformly convergent to ψ(s) as u → 1 + 0. Furthermore there exists a certain θ0 = θ0(σ1, σ2) ∈ R with 0...