Abstract
We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing new results and further reinforcing the well-known connection between Euler sums and polylogarithmic functions. Many examples of integrals of products of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given. Suggestions for further research are also suggested, including a study of polylogarithmic functions with inverse trigonometric functions.
Highlights
Introduction and PreliminariesIt is well known that integrals of products of polylogarithmic functions can be associated withEuler sums, see Reference [1]
For the case x m Lit (− x ) Liq (− x ) dx, for m ≥ 0, we show that the integral satisfies a certain recurrence relation
The work in this paper extends the results of Reference [1] and later Reference [11], in which they gave identities of products of polylogarithmic functions with positive argument in terms of zeta functions
Summary
It is well known that integrals of products of polylogarithmic functions can be associated with. This work extends the results given in Reference [1], where the author examined integrals with positive arguments of the polylogarithm. Devoto and Duke [2] list many identities of lower order polylogarithmic integrals and their relations to Euler sums. The work in this paper extends the results of Reference [1] and later Reference [11], in which they gave identities of products of polylogarithmic functions with positive argument in terms of zeta functions. Other works, including References [12,13,14,15,16,17,18,19,20,21,22,23], cite many identities of polylogarithmic integrals and Euler sums.
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