Integrable system and middle convolution

我们研究了其中一个奇异性是明显的情况下的Heun微分方程。通过对其进行中卷积，得到了一个允许多项式型解的方程，并得到了原微分方程解的积分表示。我们还研究了它与广义超几何方程的联系。我们得到了关于多指标正交多项式和超离散painlevel方程的一些结果。

We investigated Heun's differential equation in the case that one of the sigularity is apparent. By applying middle convolution for it, we obtain an equation which admits a polynomial-type solution, and we have integral representations of solutions of the original differential equation. We also studied it with a connenction to generalized hypergeometric equations.We obtained results on multi-indexed orthogonal polynomials and on ultradiscrete Painleve equations.