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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19735951 |
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- <p>We answer the open question raised in the closing remark of NGsym [Zenodo 10.5281/zenodo.19700834] on "higher-order pairs". Using the closed-form generating function established in NGFS1 [Zenodo 10.5281/zenodo.19700878], we extend the formula to G_k(z) = (1-z)^{1/z-k} prod_{l=1}^{k-1}(1-lz) for every k >= 1, and we identify the rational ratios h_k(z) = G_k(z)/G_1(z).</p> <p>Main results:<br>- Proposition (G_k formula, all k): closed form derived via partial-w differentiation of F(z,w) = (1+zw)^{1/z-1} at w = -1.<br>- Lemma (pole orders): h_k(z) has a pole of exact order k-2 at z=1 for k >= 3, with leading coefficient (-1)^{k-2}(k-2)!.<br>- Theorem (linear-extension): the only Q-linear relation sum_k alpha_k T_r(k) = 0 holding for all sufficiently large r is T_r(1) = T_r(2). Proof via Q-linear independence + transcendence of G_1.<br>- Proposition (shift-relations kernel): finite shift-relations characterised via the polynomial-coefficient map sum_k P_k(z) G_k(z).</p> <p>Pre-publication classification: DEPOSER. No mathematical bug; routine editorial polish (DOIs to references) applied before deposit.</p>