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
$\sum _{k=0}^{2m}(-1{)}^{m-k}{q}^{(\genfrac{}{}{0ex}{}{m-k}{2})}(\genfrac{}{}{0ex}{}{2m}{k})(\genfrac{}{}{0ex}{}{x+k}{2m+r})(\genfrac{}{}{0ex}{}{x+2m-k}{2m+r})$ = $\frac{q^{m(x-m-r)}(\genfrac{}{}{0ex}{}{2m}{m})}{(\genfrac{}{}{0ex}{}{2m+r}{m})}(\genfrac{}{}{0ex}{}{x}{m+r})(\genfrac{}{}{0ex}{}{x+m}{m+r})$ where $(\genfrac{}{}{0ex}{}{x}{k})$ denotes the $q$-binomial coefficient.