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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2601.04242 |
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| _version_ | 1866909984244105216 |
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| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The holonomic triangle: from a symmetry between $e$ and $π$ to additive Gamma functions Cloitre, Benoit Number Theory 11B37, 33B15, 39B32, 30E15 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. |
| title | The holonomic triangle: from a symmetry between $e$ and $π$ to additive Gamma functions |
| topic | Number Theory 11B37, 33B15, 39B32, 30E15 |
| url | https://arxiv.org/abs/2601.04242 |