  Three loop ladder and $V$-topology diagrams contributing to the massive
operator matrix element $A_{Qg}$ are calculated. The corresponding objects can
all be expressed in terms of nested sums and recurrences depending on the
Mellin variable $N$ and the dimensional parameter $\varepsilon$. Given these
representations, the desired Laurent series expansions in $\varepsilon$ can be
obtained with the help of our computer algebra toolbox. Here we rely on
generalized hypergeometric functions and Mellin-Barnes representations, on
difference ring algorithms for symbolic summation, on an optimized version of
the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on
new methods to calculate Laurent series solutions of coupled systems of
differential equations. The solutions can be computed for general coefficient
matrices directly for any basis also performing the expansion in the
dimensional parameter in case it is expressible in terms of indefinite nested
product-sum expressions. This structural result is based on new results of our
difference ring theory. In the cases discussed we deal with iterative sum- and
integral-solutions over general alphabets. The final results are expressed in
terms of special sums, forming quasi-shuffle algebras, such as nested harmonic
sums, generalized harmonic sums, and nested binomially weighted (cyclotomic)
sums. Analytic continuations to complex values of $N$ are possible through the
recursion relations obeyed by these quantities and their analytic asymptotic
expansions. The latter lead to a host of new constants beyond the multiple zeta
values, the infinite generalized harmonic and cyclotomic sums in the case of
$V$-topologies.
