Resultants, Recursive Formulas and Binet Formulas for SU(N) Verlinde Sums

Abstract

We study Verlinde sums V_n(N,m), which for our purposes are finite sums of trigonometric functions {r_x}, associated to SU(N). In this article we prove the following results: (i) the polynomial D_N(w):=∏_x(w−r_x) is equal to, up to an explicit and computable factor, the iterated resultant of Q(u)=u^m−1 with an explicit elementary polynomial W_N which is independent of n and m; (ii) for fixed N and m, V_n(N,m) satisfies a linear recursion of order at most binomm−1N−1, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of D_N; (iii) for fixed N and m, a Binet-type closed form expressing V_n(N,m) as an explicit finite sum of n-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of D_N(w) is derived. Several explicit examples are provided for N=2,3,4,5,6 and $8$.

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BibTeX RIS

BibTeX

@misc{jorgenson2026resultants,
  title = {{Resultants, Recursive Formulas and Binet Formulas for $\textrmSU(N)$ Verlinde Sums}},
  author = {Jay Jorgenson and Anders Karlsson and Lejla Smajlović},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36088},
  eprint = {2609.36088},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36088},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Resultants, Recursive Formulas and Binet Formulas for SU(N) Verlinde Sums
AU  - Jay Jorgenson
AU  - Anders Karlsson
AU  - Lejla Smajlović
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36088
UR  - https://arxiv.org/abs/2609.36088
AB  - We study Verlinde sums V_n(N,m), which for our purposes are finite sums of trigonometric functions {r_x}, associated to SU(N). In this article we prove the following results: (i) the polynomial D_N(w):=∏_x(w−r_x) is equal to, up to an explicit and computable factor, the iterated resultant of Q(u)=u^m−1 with an explicit elementary polynomial W_N which is independent of n and m; (ii) for fixed N and m, V_n(N,m) satisfies a linear recursion of order at most binomm−1N−1, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of D_N; (iii) for fixed N and m, a Binet-type closed form expressing V_n(N,m) as an explicit finite sum of n-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of D_N(w) is derived. Several explicit examples are provided for N=2,3,4,5,6 and $8$.
ER  -

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