See also Suslov (2003).
These were introduced in Carlitz (1954b, 1958). The
βn(x,q) are, in fact, rational functions of q, and not
necessarily polynomials. The Am,s(q) are always polynomials in q,
and the am,s(q) are polynomials in q for 0≤s≤m.
See §§17.8 and 20.5.