Some \(q\)-Dixon-Like Summation Formulas

Abstract

We give a \(q\)-analogue of some Dixon-like summation formulas obtained by Gould and Quaintance [Fibonacci Quart. 48 (2010), 56-61] and Chu [Integral Transforms Spec. Funct. 23 (2012), 251-261], respectively. For example, we prove that
\(\sum\limits_{k=0}^{2m} (-1)^{m-k} q^{\binom{m-k}{2}} \binom{2m} {k} \binom{x+k} {2m+r}\binom{x+2m-k} {2m+r}\) = \(\frac{q^{m(x-m-r)}\binom{2m}{m}}{\binom{2m+r}{m}}\binom{x}{m+r}\binom{x+m}{m+r}\) where \(\binom{x}{k}\) denotes the \(q\)-binomial coefficient.