394 research outputs found
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 -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
with parameters , and
\ . The sequence of generalised higher order Freud weights for
, 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
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 . We also investigate
structure and other mixed recurrence relations satisfied by the polynomials and
related properties.Comment: 17 pages, 4 figure
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