AI 中文总结
研究\(X\sim Bin(N,p)\)中与众数固定距离处二项式概率\(\Pr\{X=\nu+r\}\)，确定其关于\(N^{-1}\)幂次的完整渐近展开，通过特定结构特征组织答案，得到众数规则等结果。
AI 中文摘要
设\(X\sim Bin(N,p)\)，其中\(0<p<1\)且\(q = 1 - p\)，令\(\nu=\lceil Np\rceil\)为不低于均值的第一个格点，\(r\)为固定整数。我们确定了二项式质量\(\Pr\{X=\nu+r\}\)（一个伽马函数的商）关于\(N^{-1}\)幂次的完整渐近展开，在\((0,1)\)的紧子区间内对\(p\)以及有界的\(r\)一致成立。由于均值\(Np\)不是格点，系数不是常数，它们是在振荡分数位移\(h_N=\nu - Np\in[0,1)\)处求值的伯努利多项式，且以封闭形式给出到所有阶数。答案由三个结构特征组织起来。朴素斯特林展开的基本尾部精确求和为\(-\log\bigl((\nu+r)/(Np)\bigr)\)，并通过二项式大小偏倚恒等式去除，留下伯努利多项式中的纯Appell级数。恢复\(p\leftrightarrow q\)的对称性会将这些替换为由\(\tfrac z2\coth\tfrac z2\)生成的偶Appell序列、倒数伯努利多项式，并返回经典斯特林前置因子。整数移位\(r\)仅贡献欧拉 - 麦克劳林幂和，求和规则由生效的Appell序列固定。作为结果，\(N^{-1}\)的系数已经给出了众数的精确规则，偏度偏移为\(\tfrac12(p - q)\)；\(p=\tfrac12\)的情况恢复了中心二项式系数；并且以封闭形式获得了振荡系数的切萨罗均值。
英文摘要
We study binomial probabilities at a fixed integer distance from the upper mode, in the regime where the fractional part of the mean remains visible. The complete asymptotic expansion is obtained to all orders, uniformly for the success probability in compact subintervals of $(0,1)$ and for bounded shifts from the mode. The main point is structural. A logarithmic tail in the standard local expansion is exactly the binomial size-bias factor, and removing it leaves a pure Appell expansion. A second, symmetric normalisation is governed by an even Appell sequence and recovers the classical Stirling prefactor. The resulting coefficients explain the exact binomial mode rule, recover the central-binomial expansions as a special case, and give closed forms for averaged oscillating coefficients.
Comments22 pages
