Special Functions and River Phenomenon

Phase portrait of Ordinary Differential Equations can have special structures that strongly attract or repeal the solutions. Those structures, the “rivers”, are revealed by change of coordinates and singularly perturbed models. We first recall results for approaching this phenomenon in real and complex analysis. Then we study the connexion between rivers and special solutions of classical equations (Airy, Bessel, Kummer, Whittaker, orthogonal polynomials...) 1. River Phenomenon: a Real Analysis Approach We consider a scalar differential equation $\frac{d\mathrm{Y}}{dX}=Q(X, \mathrm{Y}\rangle (_{\backslash}\mathrm{E})$ where $Q $ is a polynomial with real coefficients: $Q(X, \mathrm{Y})=\sum_{i=1}^{p}a_{i}X^{m_{i}}Y^{n_{i}}, $ $a_{i}\in \mathbb{R} $, $(m_{i}, n_{i})\in \mathbb{Q}^{2} $. Definition 1. For equation $(E) $ , and for $r\in \mathbb{Q}$, we define the $r $-degree $ofQ(X, Y) $ to be the number $\partial_{r}Q=\max(m_{i}+rn_{i}) $. We set $\Gamma Q(X, \mathrm{Y})=