Direct Generation of a Somos-4 Sequence from an Algebraic Generating Function
Abstract
We construct a five-parameter quadratic algebraic generating function whose coefficient sequence has prescribed initial Hankel determinants $(λ,m,r,η)$ and whose Hankel transform belongs to the Somos-4 family A(1,τ). The construction starts from a Stieltjes continued fraction whose coefficients are generated by an alternating recurrence compatible with the Somos-4 relation. We derive an explicit quadratic equation for the resulting generating function and give the corresponding coefficient recurrence. A subtle feature of the construction is the nonuniqueness of a coefficient sequence determined solely by its ordinary Hankel transform. To select a canonical representative, we additionally prescribe the shifted Hankel determinants obtained from the same Somos-4 orbit advanced by two indices. This companion condition determines the odd and even Stieltjes coefficients separately and removes the remaining freedom in the continued-fraction representation. We also analyze the two algebraic branches at the origin, where they coalesce, and derive a desingularized recurrence for the coefficients. The resulting construction provides a direct algebraic generating-function realization of a general five-parameter family of Somos-4 Hankel transforms.
Related
- Algebraic characterizations of generating and affinely generating Γ-magic maps and Γ-distance magic labelings on regular graphs
- Differential Recursions, Projective Discriminants, and Algebraic Generating Functions
- Irreducibility of truncations of the Catalan generating function
- The bivariate generating function for cylindric partitions of rank three
- Algebraic Direct Sums and Generalized Drazin Invertibility
BibTeX
@misc{scheuerle2026direct,
title = {{Direct Generation of a Somos-4 Sequence from an Algebraic Generating Function}},
author = {Thomas Scheuerle},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15754},
eprint = {2609.15754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15754},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Direct Generation of a Somos-4 Sequence from an Algebraic Generating Function
AU - Thomas Scheuerle
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15754
UR - https://arxiv.org/abs/2609.15754
AB - We construct a five-parameter quadratic algebraic generating function whose coefficient sequence has prescribed initial Hankel determinants $(λ,m,r,η)$ and whose Hankel transform belongs to the Somos-4 family A(1,τ). The construction starts from a Stieltjes continued fraction whose coefficients are generated by an alternating recurrence compatible with the Somos-4 relation. We derive an explicit quadratic equation for the resulting generating function and give the corresponding coefficient recurrence. A subtle feature of the construction is the nonuniqueness of a coefficient sequence determined solely by its ordinary Hankel transform. To select a canonical representative, we additionally prescribe the shifted Hankel determinants obtained from the same Somos-4 orbit advanced by two indices. This companion condition determines the odd and even Stieltjes coefficients separately and removes the remaining freedom in the continued-fraction representation. We also analyze the two algebraic branches at the origin, where they coalesce, and derive a desingularized recurrence for the coefficients. The resulting construction provides a direct algebraic generating-function realization of a general five-parameter family of Somos-4 Hankel transforms.
ER -