PAINLEVÉ DIFFERENTIAL EQUATIONS IN THE COMPLEX PLANE (de Gruyter Studies in Mathematics 28)

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Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations (2007, methodology): “When applying Nevanlinna theory to study the growth and value distribution of meromorphic solutions of differential equations, estimates involving logarithmic derivatives have often proved to be useful [9, 22].”
Asymptotic expansions of solutions of the sixth Painlevé equation (2010, background): “For families B1, B2, only subfamilies with constant coefficients and integer exponents were known before [54].”
On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane (2006, background): “2 YIK-MAN CHIANG AND SHAO-JI FENG prime examples of integrable second order difference equations ([1], [12]).”