A common telescoping identity yields closed forms, complete asymptotics, and enveloping series for the mean absolute deviation of the four classical discrete laws.
Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion
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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.
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The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series
A common telescoping identity yields closed forms, complete asymptotics, and enveloping series for the mean absolute deviation of the four classical discrete laws.