On symmetric solutions of the fourth $q$-Painlevé equation

The Painlevé equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a $q$-difference Painlevé equation. We focus on symmetric solutions of a $q$-difference equation known as $q{\text{P}}_{\text{IV}}$ or $q\mathrm{P}(A_5^{(1)})$ and provide their symmetry properties and solve the corresponding monodromy problem.