# On the derivatives of the Heun functions

## TL;DR

This paper investigates the derivatives of Heun functions, comparing their differential equations with isomonodromy deformations related to Painlevé equations, thus advancing understanding of their mathematical properties.

## Contribution

It provides a novel comparison between the differential equations of Heun functions' derivatives and isomonodromy deformations linked to Painlevé equations.

## Key findings

- Derived differential equations for derivatives of Heun functions.
- Established connections with Painlevé equations $P_{II}-P_{VI}$.
- Enhanced understanding of the structure of Heun functions.

## Abstract

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 Painlev\'e equations $P_{II}-P_{VI}$.