Large Powers asymptotics, Khinchin families and Lagrangian distributions

TL;DR

This paper explores the use of Khinchin families combined with Local Central Limit theorems to derive new and simplified asymptotic estimates for coefficients of large powers of power series, including solutions to Lagrange's equation.

Contribution

It introduces a unified framework using Khinchin families and Local Central Limit theorems for asymptotic analysis, providing new proofs and extending classical formulas.

Findings

New proofs for asymptotic formulas of large power coefficients.

Extension of Otter and Meir-Moon asymptotic formulas.

Asymptotic results for Lagrangian probability distributions.

Abstract

This paper delves on the versatility of the theory of Khinchin families for asymptotic estimation. We show that in combination with Local Central Limit theorems for lattice variables, Khinchin families furnish a convenient and unified framework to deal with asymptotic results of the coefficients of large powers of power series. We revisit in the present paper this classical theme from that point of view, obtaining clean new proofs and a number of new results. Asymptotic results for the coefficients of solutions of Lagrange's equation fall naturally into this combined framework. We provide a direct proof of an extension of the Otter and Meir-Moon asymptotic formula as well as asymptotic results for families of Lagrangian probability distributions.