Quadratic Weyl group multiple Dirichlet series of Type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$

In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.