直接由代数生成函数生成Somos-4序列
Direct Generation of a Somos-4 Sequence from an Algebraic Generating Function
浏览论文内容
中文总结 AI 辅助
本文构造了一个五参数二次代数生成函数，通过预设初始及移位Hankel行列式消除非唯一性，直接实现了Somos-4族Hankel变换的代数生成函数表示。
中文摘要 AI 辅助
我们构造了一个五参数二次代数生成函数，其系数序列具有预设的初始Hankel行列式$(\lambda,m,r,\eta)$，且其Hankel变换属于Somos-4族$A(1,\tau)$。该构造始于一个Stieltjes连分数，其系数由与Somos-4关系相容的交替递推生成。我们推导了所得生成函数的显式二次方程，并给出了相应的系数递推。该构造的一个微妙特征在于，仅由普通Hankel变换确定的系数序列具有非唯一性。为选取一个典范代表，我们额外预设了由同一Somos-4轨道前移两个指标所得的移位Hankel行列式。这一伴随条件分别确定了奇偶Stieltjes系数，并消除了连分数表示中的剩余自由度。我们还分析了原点处的两个代数分支（它们在此处合并），并推导了系数的去奇性递推。所得构造为一般的五参数Somos-4 Hankel变换族提供了直接的代数生成函数实现。
英文摘要
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.
