Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution

Abstract

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 $\nu=\lceil Np\rceil$, 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 Ces\`aro 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.