Middle convolution and Heun's equation (Q1016686)

The author generalises his previous results [see \textit{K. Takemura}, J. Math. Anal. Appl. 342, No. 1, 52-69 (2008; Zbl 1159.34058)] concerning middle convolution of Fuchsian systems of the form \[ {{dY} \over {dz}} = \left( {{{{A_0}} \over z} + {{{A_1}} \over {z - 1}} + {{{A_t}} \over {z - t}}} \right)Y,\quad {A_0},\,{A_1},\,{A_t} \in {M_2}(C). \] Then the received new integral transformations are applied to Heun's equation \[ {{{d^2}y} \over {dz}} + \left( {{\gamma \over z} + {\delta \over {z - 1}} + {\varepsilon \over {z - t}}} \right){{dy} \over {dx}} + {{\alpha \beta z - q} \over {z(z - 1)(z - t)}}y = 0. \] It is proved, for example, that if \(\alpha \) or \(\beta \) equal to an integer \(\eta \) distinct from \(1\), the transformation \[ \begin{aligned} & y \to {y^{(\eta - 1)}},\text{{ (if }}{\kern 1pt}\eta > 1) \\ & y(z) \to \int\limits_{{C_p}} {y(w){{(z - w)}^{ - \eta }}dw} ,\text{{ (if }}{\kern 1pt}\eta < 1) \end{aligned} \] for \(p \in \{ 0,{\kern 1pt} \,1,\,{\kern 1pt} t,\,\infty \} \) gives a solution to Heun's equation (but with other parameters).