A formula for the local Heun solution

The local Heun solution is the unique solution to Heun’s equation which is analytic in the unit disk centered at $0\in \mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices.