# Classical solutions of the fifth Painlev\'e equation

AR2W02 - Mathematics of beyond all-orders phenomena

In this talk I will discuss classical solutions of the fifth Painlev\’e equation (P$$). The general solutions of the Painlev\’e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions. However, it is well known that all Painlev\’e equations except the first equation possess rational solutions, algebraic solutions and solutions expressed in terms of the classical special functions for special values of the parameters. These solutions of the Painlev\’e equations are often called classical solutions” and frequently can be expressed in the form of determinants.In the generic case of P{\rm V} when \delta\not=0, special function solutions are expressed in terms of Kummer functions and has rational solutions expressed in terms of Laguerre polynomials. In the case of P$$ when $\delta=0$, which is known as deg-P${\rm V}$ and related to the third Painlev\’e equation, special function solutions are expressed in terms of Bessel functions and has algebraic solutions expressed in terms of Laguerre polynomials. I shall give some new representations of some of these classical solutions and discuss some applications.

This talk is part of the Isaac Newton Institute Seminar Series series.