On Gromov--Witten invariants of $\mathbb{P}^1$-orbifolds and topological difference equations

Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and $\mathbb{Z}_{m_2}$ orbifold points. In this paper we introduce a matrix linear difference equation, prove existence and uniqueness of its formal Puiseux-series solutions, and use them to give conjectural formulas for $k$-point ($k\ge2$) functions of Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$. Explicit expressions of the unique solutions are also obtained. We carry out concrete computations of the first few invariants by using the conjectural formulas. For the case when one of $m_1, m_2$ equals 1, we prove validity of the conjectural formulas.