Completing the Meijer--Barnes K-function: the excluded case N<0
Abstract
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∈ℂ. There are exactly four admissible contour classes, denoted K→,K←,K↑,K↓, satisfying the universal relation K↑+K↓=K→+K← 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.
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BibTeX
@misc{he2026completing,
title = {{Completing the Meijer--Barnes $K$-function: the excluded case $N<0$}},
author = {Changhao He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16281},
eprint = {2609.16281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Completing the Meijer--Barnes K-function: the excluded case N<0
AU - Changhao He
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.16281
UR - https://arxiv.org/abs/2609.16281
AB - 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∈ℂ. There are exactly four admissible contour classes, denoted K^→,K^←,K^↑,K^↓, satisfying the universal relation K^↑+K^↓=K^→+K^← 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.
ER -