q-Extensions of identities of Abel-Rothe type

Abstract The ordinary binomial theorem may be expressed in the statement that the polynomials x n 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 a n ( x ; h , w ) = x ( x + h + nw )( x + 2 h + nw ) ⋯ ( x + ( n − 1) h + nw ), a o ( x ; h , w ) = 1, are of binomial type. When h = 0, this reduces to a symmetrized version of Abel's generalization of the binomial theorem. A different sort of generalization was made by Schutzenberger, 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 Rothe's 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.