A family of closed-form generating functions for T_r(k) in the F_p-root Laurent expansion
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2026
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| _version_ | 1866901241139822592 |
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| author | Niedbala Giraudin, David |
| author_facet | Niedbala Giraudin, David |
| contents | <p>We extend the closed-form generating function sum_r T_r(1) z^r = (1-z)^{1/z-1} to every integer k >= 1, establishing sum_{r>=0} T_r(k) z^r = (1-z)^{1/z-k} prod_{m=1}^{k-1}(1-m z), with the empty product equal to 1 at k=1. The proof is a differentiation-and-evaluation argument on the two-variable Pochhammer identity: differentiating k-1 times with respect to w and evaluating at w=-1 yields the right-hand side. As a byproduct we obtain, for each k >= 3, an exact generating-function pole of order k-2 at z=1; singularity analysis then produces the asymptotic T_r(k) ~ (-1)^{k-2} (k-2) r^{k-3}, which completes the picture across all k: the cases k=1,2 give T_r -> 0 like 1/r^2, k=3 gives a bounded T_r(3) -> -1, and k >= 4 gives polynomial growth. The family identity also yields a third, purely formal proof of the symmetry T_r(1) = T_r(2). Verifications in exact rational arithmetic through (k,r) = (5, 150) are reported.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19700904 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A family of closed-form generating functions for T_r(k) in the F_p-root Laurent expansion Niedbala Giraudin, David F_p-root distributio generating function Pochhammer symbol singularity analysis Flajolet-Sedgewick Laurent expansion Stirling numbers <p>We extend the closed-form generating function sum_r T_r(1) z^r = (1-z)^{1/z-1} to every integer k >= 1, establishing sum_{r>=0} T_r(k) z^r = (1-z)^{1/z-k} prod_{m=1}^{k-1}(1-m z), with the empty product equal to 1 at k=1. The proof is a differentiation-and-evaluation argument on the two-variable Pochhammer identity: differentiating k-1 times with respect to w and evaluating at w=-1 yields the right-hand side. As a byproduct we obtain, for each k >= 3, an exact generating-function pole of order k-2 at z=1; singularity analysis then produces the asymptotic T_r(k) ~ (-1)^{k-2} (k-2) r^{k-3}, which completes the picture across all k: the cases k=1,2 give T_r -> 0 like 1/r^2, k=3 gives a bounded T_r(3) -> -1, and k >= 4 gives polynomial growth. The family identity also yields a third, purely formal proof of the symmetry T_r(1) = T_r(2). Verifications in exact rational arithmetic through (k,r) = (5, 150) are reported.</p> |
| title | A family of closed-form generating functions for T_r(k) in the F_p-root Laurent expansion |
| topic | F_p-root distributio generating function Pochhammer symbol singularity analysis Flajolet-Sedgewick Laurent expansion Stirling numbers |
| url | https://doi.org/10.5281/zenodo.19700904 |