Classical Solutions of the Degenerate Fifth Painlev\'e Equation

In this paper classical solutions of the degenerate fifth Painlev\'e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlev\'e equation are known to expressible in terms of the third Painlev\'e equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.Comment: 19 page

An constructive proof for the Umemura polynomials for the third Painlev\'e equation

We are concerned with the Umemura polynomials associated with the third Painlev\'e equation. We extend Taneda's method, which was developed for the Yablonskii--Vorob'ev polynomials associated with the second Painlev\'e equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation satisfied by Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.Comment: 20 pages, 3 figure

Rational Solutions of the Fifth Painlev\'e Equation. Generalised Laguerre Polynomials

In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlev\'e equation, the associated $\sigma$-equation and the symmetric fifth Painlev\'e system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are determined for all values of the parameter. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.Comment: 44 pages, 22 figure

Generalised higher-order Freud weights

We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight $$\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},$$ with parameters $\lambda > -1$, $t\in\mathbb{R}$ and $m=2,3,\dots$\ . The sequence of generalised higher order Freud weights for $m=2,3,\dots$, forms a hierarchy of weights, with associated hierarchies for the first moment and the recurrence coefficient. We prove that the first moment can be written as a finite partition sum of generalised hypergeometric $_1F_m$ functions and show that the recurrence coefficients satisfy difference equations which are members of the first discrete Painlev\'e hierarchy. We analyse the asymptotic behaviour of the recurrence coefficients and the limiting distribution of the zeros as $n \to \infty$. We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.Comment: 17 pages, 4 figure