  The mirror curves enable us to study B-model topological strings on
non-compact toric Calabi--Yau threefolds. One of the method to obtain the
mirror curves is to calculate the partition function of the topological string
with single brane. In this paper, we discuss two types of geometries; one is
the chain of $N$ $\mathbb{P}^1$'s which we call `$N$-chain geometry,' the other
is the chain of $N$ $\mathbb{P}^1$'s with a compactification which we call
`periodic $N$-chain geometry.' We calculate the partition functions of the open
topological strings on these geometries, and obtain the mirror curves and their
quantization. Through the computation, we find some difference equations of
(elliptic) hypergeometric functions. We also find a relation between the
periodic chain and $\infty$-chain geometries, which implies a possible
connection between 5d and 6d gauge theories in the large $N$ limit.
