Division-based computation of higher-order asymptotic expansions

Develop a division-based technique for calculating additional terms in the asymptotic expansions of the recurrence x_{n+1}=x_n+1\pm1/x_n, whose solutions have power-logarithmic asymptotic series.

Background

Section 5 derives power-logarithmic asymptotic expansions for iterates of y/(1+y+y2) and y/(1+y-y2), equivalently for the recurrences x_{n+1}=x_n+1\pm1/x_n. The expansions involve reciprocals of power-logarithmic series.

The paper notes that extending these expansions by direct division is currently unavailable, while ordinary power-series reciprocation in related recurrences is more tractable. A systematic division-based method would permit computation of further asymptotic coefficients.

References

A division-based technique for calculating more terms in the series expansion of $x_{n}$ is not known; finding the reciprocal of a power-logarithmic series seems to be generally difficult.

— Abel's Functional Equation and Interrelations  (2503.00579 - Finch, 1 Mar 2025) in Section 5, Higher Powers