汉克尔变换与(α,β)型Somos-4序列
Hankel Transform and $(α,β)$ Somos-4 Sequences
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中文总结 AI 辅助
本文在有理函数域扩展Wang-Zhang充分条件，结合Sulanke-Xin二次变换解决Barry的(α,β)型Somos-4相关猜想，证明该序列奇偶子序列性质及汉克尔行列式相关猜想。
中文摘要 AI 辅助
(α,β)型Somos-4序列Sₙ由递推式SₙSₙ₋₄=αSₙ₋₁Sₙ₋₃+βSₙ₋₂²（n≥4）及合适的初值定义，其中α、β为常数参数。一个被广泛研究的问题是：生成函数的汉克尔变换何时会成为(α,β)型Somos-4序列？特别是，如何为这类函数推导α和β？Wang和Zhang已给出该问题的一个充分条件。本文得到三个主要结果：(i) 我们在有理函数域上扩展了Wang-Zhang充分条件，随后结合该结果与Sulanke-Xin二次变换，解决了Barry当前未解决的所有(α,β)型Somos-4猜想，这些猜想出现在广义Catalan递推、Riordan阵列、广义Bernstein阵列及椭圆曲线等不同场景中；(ii) 我们证明(α,β)型Somos-4序列的奇偶子序列仍是参数变换后的(α,β)型Somos-4序列，该结论用于证明Barry的Hurwitz变换猜想；(iii) 利用正交多项式理论，我们证明了一个汉克尔行列式公式，从而证明了一个与(α,β)型Somos-4序列相关的猜想，此外还证明了一些关于周期汉克尔行列式公式的猜想。
英文摘要
An $(α,β)$ Somos-$4$ sequence $S_n$ is defined by the recurrence $S_nS_{n-4}=αS_{n-1}S_{n-3}+βS_{n-2}^2$ ($n\geq 4$), 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.
