Hankel Transform and (α,β) Somos-4 Sequences

@misc{liu2026hankel,
  title = {{Hankel Transform and $(α,β)$ Somos-4 Sequences}},
  author = {Feihu Liu and Ying Wang and Zihao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.08703},
  eprint = {2608.08703},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08703},
  note = {CC0},
}

TY  - UNPB
TI  - Hankel Transform and (α,β) Somos-4 Sequences
AU  - Feihu Liu
AU  - Ying Wang
AU  - Zihao Zhang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.08703
UR  - https://arxiv.org/abs/2608.08703
AB  - An (α,β) Somos-$4$ sequence S_n is defined by the recurrence S_nS_n−4=αS_n−1S_n−3+βS_n−2^2 (n≥ 4), with suitable initial values, where α and β are constant parameters. A widely studied question is the following: When does the Hankel transform of a generating function become an (α,β) Somos-4 sequence? In particular, how can α and β be derived for such a function? A sufficient condition for this problem has been established by Wang and Zhang. In this paper, we obtain the following three main results. (i): We extend the Wang--Zhang sufficient condition by working over the rational function field. Then we combine this result with the Sulanke--Xin quadratic transformation to resolve all of Barry's currently unsolved (α,β) Somos-4 conjectures, which arise in diverse contexts, including generalized Catalan recurrences, Riordan arrays, generalized Bernstein arrays, and elliptic curves. (ii): We show that the odd and even subsequences of an (α,β) Somos-4 sequence are again (α,β) Somos-4 sequences with transformed parameters. This is employed to establish Barry's Hurwitz transform conjecture. (iii): Using the theory of orthogonal polynomials, we prove a Hankel determinant formula and thereby prove a conjecture related to the (α,β) Somos-4 sequence. In addition, we prove some conjectures on formulas for periodic Hankel determinants.
ER  -