Some Basic Extensions of Gustafson-Rakha's Multivariate Basic Hypergeometric Series

In this paper we extend some special cases of the multivariate basic hypergeometric series associated to the roots system of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ A_m $\end{document} that has been established and proved in [8]. For both types of the series, we will prove that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ m = 2n; n =1 $\end{document} one of the series is equivalent to Jackson's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ _8 \Psi _7 $\end{document} sum, while the other series is equivalent to the basic Gauss' sum.