Abstract :
The ordinary binomial theorem may be expressed in the statement that the polynomials xn are of binomial type. Several generalizations of the binomial theorem can be stated in this form. A particularly nice one, essentially due to Rothe, is that the polynomials an(x; h, w) = x(x + h + nw)(x + 2h + nw) ⋯ (x + (n − 1)h + nw), ao(x; h, w) = 1, are of binomial type. When h = 0, this reduces to a symmetrized version of Abelʹs generalization of the binomial theorem. A different sort of generalization was made by Schützenberger, who observed that if one adds to the statement of the binomial theorem the relation yx = qxy, then the ordinary binomial coefficient is replaced by the q-binomial coefficient. There are also commutative q-binomial theorems, one of which is subsumed in a q-Abel binomial theorem of Jackson. We go further in this direction. Our two main results are a commutative q-analogue of Rotheʹs identity with an extra parameter, and a noncommutative symmetric q-Abel identity with two extra parameters. Each of these identities contains many special cases that seem to be new.
