Some applications of the \(q\)-Rice formula (Q2772931)

The evaluation (resp. asymptotic analysis) of alternating sums involving binomial coefficients that arise e.g. as high order differences of number sequences using residue calculus has become known as Rice's method. In the present paper this basic technique consisting of finding integral representations and evaluating residues is extended to alternating sums that involve \(q\)-binomial coefficients. The central formula is NEWLINE\[NEWLINE\sum_{k=1}^n (-1)^{k-1} q^{k \choose 2} \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}} f(q^{-k}) = \frac{1}{2\pi i} \int_{\mathcal C} \frac{(q;q)_n}{(z;q)_{n+1}}f(z)dzNEWLINE\]NEWLINE where \((z;q)_n=(1-z)(1-zq)\cdots (1-zq^{n-1})\) for \(|q|<1\) and \({\mathcal C}\) is positively oriented and encircles exactly the poles \(q^{-1},q^{-2},\dots ,q^{-n}\) of the integrant. Using appropriate choices of \(f\) as some rational function the author illustrates the power of this approach by easily (re)deriving formulas due to Van Hamme (\(f(z)=\frac{1}{z-1}\)), Uchimura (\(f(z)=\frac{1}{z-q^m}\)), Dilcher (\(f(z)=\frac{1}{(z-1)^m}\)) and Andrews, Crippa and Simon (\(f(z)=\frac{z^{m-1}}{(z-1)^m}\)). The limits for \(q\to 1\) correspond to the consequences of Rice's method.