Combinatorial sums and series involving inverses of the Gaussian binomial coefficients (Q2869651)

One form of the \(q\)-binomial identity discovered by Rothe in 1811 asserts: NEWLINE\[NEWLINE(x+y)(x+qy) \cdots (x+ q^{n-1}y) = \sum_{k=0}^n {n \choose k}_q q^{\frac{k(k-1)}{2}} x^{n-k}y^kNEWLINE\]NEWLINE where \({n \choose k}_q\) is the Gaussian \(q\)-binomial coefficient. A form of the \(q\)-binomial identity can be formulated which is quite analogous to the beta-gamma identity with respect to the appropriate \(q\)-beta and \(q\)-gamma integrals. This allows a \(q\)-analogue of the expression NEWLINE\[NEWLINE\frac{1}{{n \choose k}} = (n+1) \int_0^1 t^k(1-t)^{n-k} dt.NEWLINE\]NEWLINE The author exploits this \(q\)-analogue cleverly to obtain many interesting identities for sums involving the \(q\)-binomial coefficients in the denominator.