The Liouville–Neumann approximation of the regular solutions of the Heun's equations

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We consider the standard, the confluent and the biconfluent Heun's differential equations. We apply the Liouville–Neumann expansion at a regular singular point introduced in Lopez, [J.L. Lopez, The Liouville–Neumann expansion at a regular singular point, J. Diff. Equ. Appl. 15(2) (2009), pp. 119–132] to approximate the regular solution at the origin of these three differential equations. We design, for each of the three Heun's functions, a sequence of elementary functions (rational functions and polylogarithms) that converges uniformly over compacts to the given Heun's function. This sequence (Liouville–Neumann expansion) is defined by means of an integral recurrence. Some numerical experiments show that the convergence of the Liouville–Neumann expansion is extraordinarily fast.