AI 中文总结
该研究对一类含超几何函数₅F₄的级数在特定代数点族上进行闭式求值，通过自包含证明将积分转化为对数平方和，得到含π²与对数的恒等式，完善了D'Aurizio等人的相关工作。
AI 中文摘要
我们对级数∑ₙ≥1 zⁿ/(n²C(4n,n))进行了闭式求值，该级数等于 -z/4 · ₅F₄(1,1,1,4/3,5/3;5/4,3/2,7/4,2;27z/256)，其在由有理角θ=jπ/N索引的无限代数点族上取值，每个值等于cπ²加上三个对数的通用有理二次型，其中c=-1/3(1-2j/N)²。这是D'Aurizio和Di Trani得到但未求值的四次基例，证明自包含：关联两个权重的精确整数因子结合Landen恒等式，将积分简化为对数平方和。
英文摘要
We evaluate the series $\sum_{n\ge1} z^n\big/\!\big(n^2\binom{4n}{n}\big)$, equal to $-\tfrac{z}{4}\,{}_5F_4\!\left(1,1,1,\tfrac43,\tfrac53;\tfrac54,\tfrac32,\tfrac74,2;\tfrac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $θ=jπ/N$. Each value equals $c\,π^2$ plus a universal rational quadratic form in three logarithms, with $c=-\tfrac13\left(1-\tfrac{2j}{N}\right)^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.
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