A Local Asymptotic Analysis of the First Discrete Painlev\'e Equation as the Discrete Independent Variable Approaches Infinity

The first discrete Painlev\'e equation (dPI), which appears in a model of quantum gravity, is an integrable nonlinear nonautonomous difference equation which yields the well known first Painlev\'e equation (PI) in a continuum limit. The asymptotic study of its solutions as the discrete time-step $n\to\infty$ is important both for physical application and for checking the accuracy of its role as a numerical discretization of PI. Here we show that the asymptotic analysis carried out by Boutroux (1913) for PI as its independent variable approaches infinity can also be achieved for dPI as its discrete independent variable approaches the same limit.