Search

The connection between polylogarithmic functions and Euler sums is well known. In this paper we explore the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider mainly, polylogarithmic functions with negative arguments, thereby producing new results and extending the work of Freitas. Many examples of integrals of products of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given.

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.

An investigation into families of integrals containing powers of the arctanh and log functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals. In another special case we prove that the corresponding log-tanh integral can be represented as a linear combination of the product of zeta functions and the Dirichlet eta function. [ABSTRACT FROM AUTHOR]

We evaluate Euler-like sums involving harmonic numbers using expansions for powers of logarithmic, arctan and a r c t a n h functions. [ABSTRACT FROM AUTHOR]

In this paper we develop new identities for Euler sums with rational argument. We shall investigate and report on new Euler identities of weight for m,p,q positive integers and m + p + q an even integer, but with a non-unitary argument of the harmonic numbers. Some examples of these Euler identities will be given in terms of Riemann zeta values, Dirichlet values and other special functions. We shall also give an explicit solution to an integral with a log polylog integrand. [ABSTRACT FROM AUTHOR]

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series. [ABSTRACT FROM AUTHOR]

We develop families of generalized hypergeometric function identities at argument±1 in terms of generalized harmonic numbers and Hurwitz zeta functions. [ABSTRACT FROM AUTHOR]