  The six-vertex model with Domain Wall Boundary Conditions, or square ice, is
considered for particular values of its parameters, corresponding to 1-, 2-,
and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel
determinant representations for the partition function and the boundary
correlator of homogeneous square ice, it is shown how the ordinary and refined
enumerations can be derived in a very simple and straightforward way. The
derivation is based on the standard relationship between Hankel determinants
and orthogonal polynomials. For the particular sets of parameters corresponding
to 1-, 2-, and 3-enumerations of ASMs, the Hankel determinant can be naturally
related to Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn
polynomials, respectively. This observation allows for a unified and simplified
treatment of ASMs enumerations. In particular, along the lines of the proposed
approach, we provide a complete solution to the long standing problem of the
refined 3-enumeration of AMSs.
