Asymptotics of bivariate algebraico-logarithmic generating functions

TL;DR

This paper derives asymptotic formulas for coefficients of bivariate generating functions with algebraic and logarithmic factors, aiding combinatorial analysis and random object generation.

Contribution

It extends contour manipulation methods to D-finite generating functions with algebraic-logarithmic factors, providing quickly computable asymptotics.

Findings

Asymptotic formulas for coefficients are derived.

Method applicable to D-finite functions with logarithmic factors.

Facilitates verification of combinatorial properties and random generation.

Abstract

We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken into indecomposable parts. Asymptotics are quickly computable and can verify combinatorial properties of sequences and assist in randomly generating objects. While multiple approaches for algebraic asymptotics have recently emerged, we find that the contour manipulation approach can be extended to these D-finite generating functions.