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The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one more singularity. In this paper we compare these equations with linear differential equations isomonodromy deformations of which are described by the Painlevé equations \(P_{II}-P_{VI}\).

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We thank M. Nieszporski (University of Warsaw) for interesting discussions. G.F. acknowledges the support of the National Science Center (Poland) via grant OPUS 2017/25/ B/BST1/00931 and the Alexander von Humboldt Foundation. The support of the Armenian State Committee of Science (SCS grants no. 18RF-139 and No. 18T-1C276), the Armenian National Science and Education Fund (ANSEF grant no. PS-4986), the Russian-Armenian (Slavonic) University is also greatfully acknowledged. A.I. thanks the colleagues from the University of Warsaw for hospitality and inspiring discussions.

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Filipuk, G., Ishkhanyan, A. & Dereziński, J. On the Derivatives of the Heun Functions. J. Contemp. Mathemat. Anal. 55, 200–207 (2020).

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