Sum formulas for double polylogarithms with a shifting parameter and their derivatives

TL;DR

This paper establishes new sum formulas for double polylogarithms with a shifting parameter, connecting them to known zeta and L-value formulas, and explores their derivatives and partial sums.

Contribution

It introduces sum formulas for Hurwitz-type double polylogarithms with a shifting parameter and derives related weighted sum formulas through differentiation.

Findings

Sum formulas for double polylogarithms with a shifting parameter

Weighted sum formulas obtained via differentiation in the parameter

Sum formulas for partial double zeta values with congruence conditions

Abstract

We prove sum formulas for double polylogarithms of Hurwitz type, that is, involving a shifting parameter $b$ in the denominator. These formulas especially imply well-known sum formulas for double zeta values, and sum formulas for double $L$-values. Further, differentiating in $b$, we obtain a kind of weighted sum formula for double polylogarithms and double $L$-values. We also give sum formulas for partial double zeta values with some congruence conditions. Our proofs of those sum formulas are based on certain functional relations for double polylogarithms of Hurwitz type.