  We analyze the effect of a simple coin operator, built out of Bell pairs, in
a $2d$ Discrete Quantum Random Walk (DQRW) problem. The specific form of the
coin enables us to find analytical and closed form solutions to the recursion
relations of the DQRW. The coin induces entanglement between the spin and
position degrees of freedom, which oscillates with time and reaches a constant
value asymptotically. We probe the entangling properties of the coin operator
further, by two different measures. First, by integrating over the space of
initial tensor product states, we determine the {\it Entangling Power} of the
coin operator. Secondly, we compute the {\it Generalized Relative R\'{e}nyi
Entropy} between the corresponding density matrices for the entangled state and
the initial pure unentangled state. Both the {\it Entangling Power} and {\it
Generalized Relative R\'{e}nyi Entropy} behaves similar to the entanglement
with time. Finally, in the continuum limit, the specific coin operator reduces
the $2d$ DQRW into two $1d$ massive fermions coupled to synthetic gauge fields,
where both the mass term and the gauge fields are built out of the coin
parameters.
