Asymptotic Expansions of the Reciprocal Beta Integral via Tricomi–Erdélyi Coefficients

Abstract

The analytical properties of the reciprocal Beta function and its discrete counterpart, the reciprocals of binomial coefficients, have long been a focal point in combinatorics and classical asymptotic analysis. While existing literature primarily concentrates on discrete summation identities over the integer domain, this paper bridges the gap between discrete and continuous frameworks by systematically investigating the continuous integral of the reciprocal Beta function. Motivated by a classical combinatorial problem concerning the relative ordering of elements in uniform random permutations, we address the finite integral , which diverges as the upper limit . We establish that this divergent behavior can be elegantly regularized and decoded. Specifically, when scaled by a prefactor , the expression permits a full asymptotic expansion as , which can be uniquely expressed as an infinite Laurent series in inverse powers of : where each subleading coefficient is a rational function of the parameter and completely independent of . By performing a rigorous coefficient matching via the classical Tricomi–Erdélyi asymptotic expansion framework and implementing symbolic computer algebra, we explicitly determine the exact closed-form expressions for the leading constant and the higher-order subleading coefficients up to . Our results refine the understanding of continuous Beta-like integrals and expand the modern toolkit of hyperasymptotic expansions for ratios of special functions.