example with length k = 2 and weight n = 7: Euler sum with all possible alternations of sign The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum. Notation with are the generalized harmonic numbers. with with with As a variant of the Dirichlet eta function we define with Reflection formula The reflection formula can be generalized as follows: if we have Other relations Using the series definition it is easy to prove: with with A further useful relation is: where and Note that must be used for all value for which the argument of the factorials is Other results For all positive integers : or more generally: Mordell–Tornheim zeta values The Mordell–Tornheim zeta function, introduced by who was motivated by the papers and , is defined by It is a special case of the Shintani zeta function. References Notes External links Zeta and L-functions