  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 in analyzing lower order terms in the behavior of zeros
of L-functions near the central point.
