AI 中文总结
本研究针对实Rogers双对数构造单参数函数方程，通过Rogers五项关系证明，得到具有四个实根的全实四次双对数梯子，为相关领域提供了新的恒等式与梯子结果。
AI 中文摘要
我们针对实Rogers双对数给出一个单参数函数方程，其自变量属于 $\boldsymbol{\bigl(u,\boldsymbol{\bigr)}}$，并通过Rogers五项关系的十个显式实例证明该方程：由此产生的五十个自变量抵消后得到恒等式的五个自变量，且这五个自变量之间不存在更短的关系。该方程来自一对初等相等的积分，我们证明其基础被积函数是本质上被强制的。对参数进行特化可得到 $\boldsymbol{\bigl(\boldsymbol{\bigr)}}$ 和 $\boldsymbol{\bigl(\boldsymbol{\bigr)}}$ 上的恒等式，涉及 $\boldsymbol{\bigl(\boldsymbol{\bigr)}}$、$\boldsymbol{\bigl(\boldsymbol{\bigr)}}$ 与 $\boldsymbol{\bigl(\boldsymbol{\bigr)}}$ 的关系，还得到Catalan常数的新类似物，以及 $\boldsymbol{\bigl(\boldsymbol{\bigr)}}$ 上的一对四次基双对数梯子。这些梯子的基方程和整数关系搜索得到的四个推测梯子的基方程都是具有四个实根的不可约四次方程。据我们所知，之前记录的所有四次双对数梯子的基方程都只有两个实根，因此这些似乎是首批次数超过三的全实双对数梯子。
英文摘要
We give a one-parameter functional equation for the real Rogers dilogarithm, with arguments in $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$, and prove it by an explicit array of ten instances of Rogers' five-term relation: the fifty arguments so contributed cancel down to the five of the identity, which admit no shorter relation among themselves. The equation comes from a pair of integrals whose equality is elementary, and we show that the underlying integrand is essentially forced. Specialising the parameter gives identities over $\mathbb{Q}(\sqrt{33})$ and $\mathbb{Q}(\sqrt{17})$, a relation between $\mathbb{Q}(\sqrt{13})$, $\mathbb{Q}(\sqrt{3})$, and $\operatorname{Cl}_2(π/3)$ with a new analogue for Catalan's constant, and a pair of dilogarithm ladders of quartic base over $\mathbb{Q}(\sqrt{33})$. The base equations of these ladders, and of four conjectural ones located by integer-relation search, are irreducible quartics with four real roots. All previously recorded ladders of degree four known to us have base equations with two real roots, so these appear to be the first totally real ladders of degree exceeding three.
Comments22 pages
