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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.04242 |
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Table of Contents:
- Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $π$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational coefficients. We call such functions additive Gamma functions (AGFs), recognizing Euler's $Γ(z)$ as the order-1 prototype. Our theory reveals a structural dichotomy: one AGF is expressible as Gamma ratios (regular case), another involves incomplete Gamma (irregular case). AGFs complete a holonomic triangle between P-recursive sequences, additive functional equations, and differential equations, unifying discrete and continuous perspectives under the condition that Gamma factors in asymptotics have integer slopes.