AI 中文总结
该研究构造三参数有理逼近，证明拉马努金θ函数等三类q级数在特定点的值为无理数，给出其无理性测度上界，并发现其中一个逼近与Coussement–Smet的帕德逼近一致。
AI 中文摘要
我们构造了一个三参数族的有理逼近，用于逼近q超几何级数的值。利用这些逼近，我们证明了对于每个满足|x|≥2的整数x，拉马努金θ函数ψ(r)=∑_{n≥0}r^{n(n+1)/2}在r=x⁻¹处的值、奇整数上的除数函数生成函数Δ(r)=∑_{m≥0}d(2m+1)r^m，以及4-正则分拆的生成函数B₄(r)=∑_{n≥0}b₄(n)r^m均为无理数。我们还分别得到它们的无理性测度上界为18/7、18π²/(7π²−24)和3。此外，我们证明其中一个构造的逼近与Coussement–Smet提出的兰伯特级数的帕德逼近一致。
英文摘要
We construct a three-parameter family of rational approximations to values of $q$-hypergeometric series. Using these approximations, we prove that, for every integer $x$ with $|x|\geq2$, the values at $r=x^{-1}$ of Ramanujan's theta function $ψ(r)=\sum_{n\geq0}r^{n(n+1)/2}$, the generating function $Δ(r)=\sum_{m\geq0}d(2m+1)r^m$ of the divisor function restricted to odd integers, and the generating function $B_4(r)=\sum_{n\geq0}b_4(n)r^n$ for $4$-regular partitions are irrational. We further obtain the upper bounds $18/7$, $18π^2/(7π^2-24)$, and $3$, respectively, for their irrationality measures. We also show that one of the constructed approximations coincides with the Padé approximation to a Lambert series due to Coussement--Smet.
Comments37 pages. Comments welcome!
