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