发表机构
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文补全了Meijer--Barnes $K$-函数在 $N<0$ 情形下的定义，证明了积分收敛性，刻画了四类容许围道及其关系，推导了留数展开和变换律，并提供了数值验证。
AI 中文摘要
Karp和Kuznetsov通过将Meijer $G$-函数的Mellin--Barnes核中的Euler gamma因子替换为Barnes双gamma因子，引入了Meijer--Barnes $K$-函数，但排除了 $N=2(m+n)-p-q<0$ 的情形。我们完成了这一缺失情形。该核在两个水平扇形区域内超指数衰减，定义积分在对数黎曼曲面上的每个 $z$ 和每个 $\alpha\in\mathbb{C}$ 上绝对收敛。恰好有四类容许围道，记为 $K^\rightarrow,K^\leftarrow,K^\uparrow,K^\downarrow$，满足普适关系 $K^\uparrow+K^\downarrow=K^\rightarrow+K^\leftarrow$，因此张成至多三维的空间。在显式的闭弧增长假设下，我们推导了留数展开和加强的消失定理；三个基本变换律无条件成立。我们还证明了对于 $N<0$ 不存在垂直围道是容许的，用相容的两指标递推关系替代标准的垂直围道Mellin变换方法，并发展了核级插入演算以及解析变换恒等式，其中所需的交换是合理的，包括整数阶和Riemann--Liouville分数阶导数。最后，我们为一般 $\tau>0$ 的Barnes双gamma函数构建了一个数值计算器，并高精度验证了代表性恒等式和围道公式。
英文摘要
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 $α\in\mathbb{C}$. 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 $τ>0$ and validate representative identities and contour formulas to high precision.
Comments32 pages, 3 figures, 4 tables. Numerical validation scripts are included in the source package
