Discrete Polylogarithm Functions

An Identity for Sums of Polylogarithm Functions

We derive an identity for certain linear combinations of polylogarithm functions with negative exponents, which implies relations for linear combinations of Eulerian numbers. The coefficients of our linear combinations are related to expanding moments of Satake parameters of holomorphic cuspidal newforms in terms of the moments of the corresponding Fourier coefficients, which has applications i...

IDENTITIES FOR SUMS OF A q-ANALOGUE OF POLYLOGARITHM FUNCTIONS

We present several identities for sums of q-polylogarithm functions. Our motivation for these are the relations between the q-zeta function (see [3, 4, 12] and references therein), q-polylogarithm functions (see below) and the quantum group SUq(2). More precisely, the left regular representation of the quantum group SUq(2) is the coordinate ring A(SUq(2)) represented as a left Uq(sl(2))-module....

Tempering the Polylogarithm

converges when |z| < 1, defining the classical polylogarithm function, equal to − log(1 − z) when s = 1. In general, the behavior of these functions at z = 1 is complicated: it is known, for example, that Lis has an analytic continuation to the cut plane C − 1,∞). The fact that the ‘modified’ polylogarithm of Bloch, Ramakrishnan, Wigner, Wojtkowiak, Zagier, . . . , defined as the real, or imagi...

A Sixteenth-order Polylogarithm Ladder

ZETA SERIES GENERATING FUNCTION TRANSFORMATIONS RELATED TO POLYLOGARITHM FUNCTIONS AND THE k-ORDER HARMONIC NUMBERS

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the jth derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a n...