Completing the Meijer--Barnes $K$-function: the excluded case $N<0$

RESUMO

Karp and Kuznetsov introduced the Meijer--Barnes $K$-function by replacing the Euler gamma factors in the Mellin--Barnes kernel of the Meijer $G$-function with Barnes double gamma factors, but excluded the regime $N=2(m+n)-p-q<0$. We complete this missing case. The kernel decays super-exponentially in two horizontal sectors, and the defining integral converges absolutely for every $z$ on the Riemann surface of the logarithm and every $\alpha\in\mathbb{C}$. There are exactly four admissible contour classes, denoted $K^\rightarrow,K^\leftarrow,K^\uparrow,K^\downarrow$, satisfying the universal relation $K^\uparrow+K^\downarrow=K^\rightarrow+K^\leftarrow$ and hence spanning a space of dimension at most three. Under an explicit closing-arc growth hypothesis, we derive residue expansions and strengthened vanishing theorems; the three basic transformation laws hold unconditionally. We also prove that no vertical contour is admissible for $N<0$, replace the standard vertical-contour Mellin-transform method by a compatible two-index recurrence for residue coefficients, and develop a kernel-level insertion calculus together with analytic transform identities where the required interchange is justified, including integer and Riemann--Liouville fractional derivatives. Finally, we construct a numerical evaluator for the Barnes double gamma function at general $\tau>0$ and validate representative identities and contour formulas to high precision.