q-Extensions of identities of Abel-Rothe type

✍ Scribed by Warren P. Johnson

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 629 KB
• Volume: 159
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(95)00108-9

✦ Synopsis

The ordinary binomial theorem may be expressed in the statement that the polynomials x" 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,(x; h,w)=

x(x+h+nw)(x+2h+nw)...(x+(n-1)h+nw),

ao(x; h, w) = l, 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 Schiitzenberger, 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.