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arXiv · 2212.11513

On symmetric solutions of the fourth $q$-Painlev\'e equation

Nalini Joshi·Pieter Roffelsen
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Abstract

The Painlev\'e 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 Painlev\'e equation. We focus on symmetric solutions of a $q$-difference equation known as $q\textrm{P}_{\textrm{IV}}$ or $q{\rm P}(A_5^{(1)})$ and provide their symmetry properties and solve the corresponding monodromy problem.

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Nalini Joshi, Pieter Roffelsen. 2022-12-22. On symmetric solutions of the fourth $q$-Painlev\'e equation. https://doi.org/10.1088/1751-8121/acc7dc

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