The reconstruction of linear ODEs from their monodromy leads to two interesting classes of special functions: Heun accessory parameters (for the simplest 2nd order scalar ODEs) and Painlevé functions (for the simplest 1st order 2 by 2 linear systems). I will discuss several approaches to computation of these functions, such as Hill and Widom determinant, continued fractions and combinatorial series. After recalling a classical relation between Heun and Painlevé equations, I will explain how it leads to identities expressing Heun accessory parameters in terms of Painlevé functions and vice versa. Speaker(s): Oleg Lisovyi (Université de Tours)