  Quantum walks are known to have nontrivial interaction with absorbing
boundaries. In particular, Ambainis et.\ al.\ \cite{ambainis01} showed that in
the $(\Z ,C_1,H)$ quantum walk (one-dimensional Hadamard walk) an absorbing
boundary partially reflects information. These authors also conjectured that
the left absorption probabilities $P_n^{(1)}(1,0)$ related to the finite
absorbing Hadamard walks $(\Z ,C_1,H,\{ 0,n\} )$ satisfy a linear fractional
recurrence in $n$ (here $P_n(1,0)$ is the probability that a Hadamard walk
particle initialized in $|1\rangle |R\rangle$ is eventually absorbed at
$|0\rangle$ and not at $|n\rangle$). This result, as well as a third order
linear recurrence in initial position $m$ of $P_n^{(m)}(1,0)$, was later proved
by Bach and Borisov \cite{bach09} using techniques from complex analysis. In
this paper we extend these results to general two state quantum walks and
three-state Grover walks, while providing a partial calculation for absorption
in $d$-dimensional Grover walks by a $d-1$-dimensional wall. In the
one-dimensional cases, we prove partial reflection of information, a linear
fractional recurrence in lattice size, and a linear recurrence in initial
position.
