  We consider a class of sums over products of Z-sums whose arguments differ by
a symbolic integer. Such sums appear, for instance, in the expansion of Gauss
hypergeometric functions around integer indices that depend on a symbolic
parameter. We present a telescopic algorithm for efficiently converting these
sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for
generic values of this parameter. This algorithm is illustrated by computing
the double pentaladder integrals through ten loops, and a family of massive
self-energy diagrams through $O(\epsilon^6)$ in dimensional regularization. We
also outline the general telescopic strategy of this algorithm, which we
anticipate can be applied to other classes of sums.
