We derive new functional equations for Nielsen polylogarithms. show that, when viewed moduloLi5 and products of lower weight functions, the 5 polylogarithm S3,2 satisfies dilogarithm five-term relation. also give some evaluations polylogarithms in weights up to 8, general families identities higher weight.

The finite logarithm was introduced by Kontsevich (under the name “The 1 1 2 logarithm”) in [Kon]. The finite logarithm is the case n = 1 of the n-th polylogarithm lin ∈ Z/p[z] defined by lin(z) = ∑p−1 k=1 z /k. In loc. cit. Kontsevich proved that the finite logarithm satisfies a 4-term functional equation, known as the fundamental equation of information theory. The same functional equation is...

We first study some generalizations of Eulerian fractions with complex order parameter and investigate their interrelationship likewise generalized functions as well Stirling functions. apply the new approach to polylogarithms non-integral order, for which only a few values are known in closed form. In particular, we present structural solution counterpart an old conjecture Mengoli Euler polylo...

In this work we study the behavior of relativistic ideal Bose and Fermi gases in two space dimensions. Making use of polylogarithm functions we derive a closed and unified expression for their densities. It is shown that both type of gases are essentially inequivalent, and only in the non-relativistic limit the spinless and equal mass Bose and Fermi gases are equivalent as known in the literature.

In this paper we construct trigonometric functions of the sum Tp(h, k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized La...

We give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms. As an application, we generalize the well-known decomposition formula of Euler that expresses the product of two Riemann zeta values as a sum of double zeta values to a formula that expresses the product of t...

In this work we study the behavior of relativistic ideal Bose and Fermi gases in two space dimensions. Making use of polylogarithm functions we derive a closed and unified expression for their densities. It is shown that both type of gases are essentially inequivalent, and only in the non-relativistic limit the spinless and equal mass Bose and Fermi gases are equivalent as known in the literature.

We introduce the poly-Cauchy polynomials which generalize the classical Cauchy polynomials, and investigate their arithmetical and combinatorial properties. These polynomials are considered as analogues of the poly-Bernoulli polynomials that generalize the classical Bernoulli polynomials. Moreover, we investigate the zeta functions which interpolate the poly-Cauchy polynomials. The values of th...

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot 1 theory and high-energy physics. More recently, we have been forced to consider mul-tidimensional extensions encompassing the classical polylogarit...