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

Abstract

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.

27 pages, 7 figures. Parts of the text on the discrete symmetries ${\mathcal{T}}_{\pm }$ and their relation to the symmetry group updated, emphasising that both correspond to one and the same Dynkin diagram automorphism. The title of section 5 has been changed, typos have been fixed and other minor updates

References in corpus (2)

• Poles of Painlevé IV Rationals and their Distribution
• Special Solutions of the Sixth Painleve Equation with Solvable Monodromy