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SUMMARY:Classical solutions of the fifth Painlev\\'e equation - Peter Clar
 kson (University of Kent)
DTSTART:20221103T100000Z
DTEND:20221103T105000Z
UID:TALK185225@talks.cam.ac.uk
DESCRIPTION:In this talk I will discuss classical solutions of the fifth P
 ainlev\\'e equation (P$_{\\rm V}$).&nbsp\;The general solutions of the Pai
 nlev\\'e equations are transcendental in the sense that they cannot be exp
 ressed 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 cl
 assical special functions for special values of the parameters. These solu
 tions of the Painlev\\'e equations are often called ``classical solutions"
  and frequently can be expressed in the form of determinants.In the generi
 c case of P$_{\\rm V}$ when $\\delta\\not=0$\, special function solutions 
 are expressed in terms of Kummer functions and has rational solutions expr
 essed in terms of Laguerre polynomials. In the case of P$_{\\rm V}$ when $
 \\delta=0$\, which is known as deg-P$_{\\rm V}$ and related to the third P
 ainlev\\'e equation\, special function solutions are expressed in terms of
  Bessel functions and has algebraic solutions expressed in terms of Laguer
 re polynomials. I shall give some new representations of some of these cla
 ssical solutions and discuss some applications.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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