Linear recurrences for S_r(k) in the F_p-root Laurent expansion
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| Natura: | Recurso digital |
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2026
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| _version_ | 1866901257053011968 |
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| author | Niedbala Giraudin, David |
| author_facet | Niedbala Giraudin, David |
| contents | <p>Starting from the closed-form generating functions sum_r T_r(k) z^r = (1-z)^{1/z-k} prod_{m=1}^{k-1}(1-mz), we derive the linear recurrence r S_r(k) = sum_{j=1}^r c_j^{(k)} S_{r-j}(k) with S_0(k) = 1 and coefficients c_j^{(k)} = 1/(j+1) + (k-1) - sum_{m=1}^{k-1} m^j. The identity c_j^{(1)} = c_j^{(2)} = 1/(j+1) yields a fourth, purely algebraic proof of the symmetry S_r(1) = S_r(2). The recurrence is verified in exact rational arithmetic for k in {1, 2, 3, 4, 5} and r in {0, ..., 12} and provides an O(r^2)-time algorithm for computing S_r(k), compared to O(r^3) for the Vandermonde/falling-factorial method.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19700972 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Linear recurrences for S_r(k) in the F_p-root Laurent expansion Niedbala Giraudin, David F_p-root distribution linear recurrence generating function Bernoulli polynomials power sums Laurent expansion <p>Starting from the closed-form generating functions sum_r T_r(k) z^r = (1-z)^{1/z-k} prod_{m=1}^{k-1}(1-mz), we derive the linear recurrence r S_r(k) = sum_{j=1}^r c_j^{(k)} S_{r-j}(k) with S_0(k) = 1 and coefficients c_j^{(k)} = 1/(j+1) + (k-1) - sum_{m=1}^{k-1} m^j. The identity c_j^{(1)} = c_j^{(2)} = 1/(j+1) yields a fourth, purely algebraic proof of the symmetry S_r(1) = S_r(2). The recurrence is verified in exact rational arithmetic for k in {1, 2, 3, 4, 5} and r in {0, ..., 12} and provides an O(r^2)-time algorithm for computing S_r(k), compared to O(r^3) for the Vandermonde/falling-factorial method.</p> |
| title | Linear recurrences for S_r(k) in the F_p-root Laurent expansion |
| topic | F_p-root distribution linear recurrence generating function Bernoulli polynomials power sums Laurent expansion |
| url | https://doi.org/10.5281/zenodo.19700972 |