q-Extensions of identities of Abel-Rothe type Original Research Article

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.