四亏格Kronecker极限公式对交错Rogers-Ramanujan连分数的求值
A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction
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中文总结 AI 辅助
本文用Kronecker极限公式的亏格-特征形式，对六个类数四的奇数情形求值交错Rogers-Ramanujan连分数，结果以实二次域基本单位及根式表达，并附数值验证。
中文摘要 AI 辅助
我们求值了Ramanathan处理Rogers-Ramanujan连分数时遗留的六个奇数类数四的情形。设\\[ S(q)=-R(-q),\qquad R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}. \\] 对于$n=39,87,111,119,159,287$，我们通过将Kronecker极限公式的亏格-特征形式应用于$\mathbb Q(\sqrt{-5n})$的四个理想类来确定$S(e^{-\pi/\sqrt{5n}})$的值。所得表达式以实二次域的基本单位给出。特别地，若$X_n=S(e^{-\pi/\sqrt{5n}})$，则\\[ X_n^{-5}+11-X_n^5=\frac{5\sqrt5}{U_n}, \\] 其中六个量$U_n$在下面明确展示。我们还给出了相应的根式表达式和四次代数证书，以及六个求值的数值检验。
英文摘要
We evaluate the six odd class-number-four cases left unevaluated in Ramanathan's treatment of the Rogers-Ramanujan continued fraction. Let \[ S(q)=-R(-q),\qquad R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}. \] For $n=39,87,111,119,159,287$ we determine the value of $S(e^{-π/\sqrt{5n}})$ by applying the genus-character form of the Kronecker limit formula to the four ideal classes of $\mathbb Q(\sqrt{-5n})$. The resulting expressions are given in terms of fundamental units of real quadratic fields. In particular, if $X_n=S(e^{-π/\sqrt{5n}})$, then \[ X_n^{-5}+11-X_n^5=\frac{5\sqrt5}{U_n}, \] where the six quantities $U_n$ are displayed explicitly below. We also give the corresponding radical expressions and quartic algebraic certificates, together with numerical checks of the six evaluations.
