AI 中文总结
研究二项分布在有界晶格位移处的质量及平均绝对偏差的渐近展开，通过伽马商展开等方法，得到由伯努利多项式控制的展开式，还得出相关系数性质及边界，如切萨罗均值、对数双边界等。
AI 中文摘要
我们推导了在有界晶格位移处二项式质量以及平均绝对偏差\(E|X - Np|\)（其中\(X\sim Bin(N,p)\)，\(0 < p < 1\)）的完整渐近展开式。棣莫弗的精确公式将后一个问题简化为\(\nu=\lceil Np\rceil\)处的局部质量，所以系数取决于振荡位移\(h_N=\lceil Np\rceil - Np\)。我们表明完整展开式由在此位移处求值的伯努利多项式控制；等价地，晶格校正为斯特林级数中的阿佩尔移位。计算基于具有不等线性缩放的伽马商展开式，并在有界移位中具有一致性。从局部质量到平均绝对偏差时，移位的基本非伯努利部分与棣莫弗预因子逐项抵消，留下纯伯努利多项式系数。结果，Frame和Johnson的经典一阶校正嵌入在一般系数序列中，并且振荡系数的切萨罗均值可从伯努利多项式的乘法定理获得。最后，虽然振荡渐近展开式不能通过截断来界定其函数，但棣莫弗恒等式与罗宾斯形式的斯特林公式在内部产生了对数宽度为\(O(N^{-2})\)的基本双边界；在整数均值处，对数展开式简化为\(N^{-1}\)奇次幂的符号交替级数，我们通过组合三伽马斯特林余项的符号确定比内核表示证明其是包络的：连续截断将平均绝对偏差括在中间。
英文摘要
We derive complete asymptotic expansions for the binomial mass at a bounded lattice displacement and for the mean absolute deviation $E|X-Np|$, $X\sim Bin(N,p)$, with $0<p<1$. De Moivre's exact formula reduces the latter problem to the local mass at $ν=\lceil Np\rceil$, so the coefficients depend on the oscillating displacement $h_N=\lceil Np\rceil-Np$. We show that the full expansion is governed by Bernoulli polynomials evaluated at this displacement; equivalently, the lattice correction is an Appell shift in the Stirling series. The calculation is based on a gamma-quotient expansion with unequal linear scalings, stated with uniformity in the bounded shift. In passing from the local mass to the mean absolute deviation, the elementary, non-Bernoulli part of the shift cancels term by term against the De Moivre prefactor, leaving coefficients that are pure Bernoulli polynomials. As consequences, the classical first correction of Frame and Johnson is embedded in the general coefficient sequence, and the Cesàro means of the oscillating coefficients are obtained from the multiplication theorem for Bernoulli polynomials. Finally, although an oscillating asymptotic expansion does not bound its function by truncation, De Moivre's identity together with Robbins's form of Stirling's formula yields an elementary two-sided bound of logarithmic width $O(N^{-2})$ in the interior; and at an integer mean the logarithmic expansion reduces to a sign-alternating series in odd powers of $N^{-1}$ which we prove to be enveloping, via a sign-definite Binet-kernel representation of the combined three-gamma Stirling remainder: successive truncations bracket the mean absolute deviation.
Comments18 pages
