AI 中文总结
本文研究格雷戈里系数的整插值函数\\(\mathcal{G}(z)\\)，推导其完全形式满足的马尔可夫变换恒等式，分析其零点性质，得到\\(\rho_n\\)的渐近式等结果，给出了欧拉常数\\(\gamma\\)的新表达式。
AI 中文摘要
我们研究格雷戈里系数的整插值函数\\(\mathcal{G}(z)=\int_0^1 \binom{x}{z}\\,dx\\)，其完全形式满足正马尔可夫变换恒等式\\(\frac{\pi z}{\sin(\pi z)}\mathcal{G}(z) =\sum_{n=1}^{\infty}\frac{n\left|G_n\right|}{n-z}\\)。由此可得，\\(\mathcal{G}(z)\\)的所有零点均为实单零点，负零点为整数\\(-1,-2,\ldots\\)，且每个区间\\((n,n+1)\\)内存在一个零点\\(\rho_n\\)。我们推导了\\(\rho_n-n\\)的完全对数渐近式，确定了\\(\mathcal{G}\\)的卡特赖特增长性与典型乘积，实现了\\(1/\rho_n\\)的谱表示，所得相对行列式给出\\(\gamma=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{\rho_n}\right)\\)。
英文摘要
We study the entire interpolation \[ \mathcal{G}(z)=\int_0^1 \binom{x}{z}\,dx \] of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity \[ \frac{πz}{\sin(πz)}\mathcal{G}(z) =\sum_{n=1}^{\infty}\frac{n\left|G_n\right|}{n-z}. \] Consequently, every zero is real and simple; the negative zeros are the integers $-1,-2,\ldots$, and one zero $ρ_n$ lies in each $(n,n+1)$. We derive complete logarithmic asymptotics for $ρ_n-n$, determine the Cartwright growth and canonical products of $\mathcal{G}$, and realize $1/ρ_n$ spectrally. The resulting relative determinant yields \[ γ=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{ρ_n}\right). \]
Comments29 pages
