  It is shown that an extended q-deformed $su(2)$ algebra with an extra
(``Schwinger '') term can describe Bloch electrons in a uniform magnetic field
with an additional periodic potential. This is a generalization of the analysis
of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory
of this q-deformed algebra, we obtain functional Bethe ansatz equations whose
solutions should be functions of finite degree. It is also shown that the zero
energy solution is expressed in terms of an Askey Wilson polynomial.
