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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Classical solutions of the fifth Painlev\\'e equat
ion - Peter Clarkson (University of Kent)
DTSTART;TZID=Europe/London:20221103T100000
DTEND;TZID=Europe/London:20221103T105000
UID:TALK185225AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/185225
DESCRIPTION:In this talk I will discuss classical solutions of
the fifth Painlev\\'e equation (P$_{\\rm V}$).&nb
sp\;The general solutions of the Painlev\\'e equat
ions are transcendental in the sense that they can
not be expressed in terms of known elementary func
tions. However\, it is well known that all Painlev
\\'e equations except the first equation possess r
ational solutions\, algebraic solutions and soluti
ons expressed in terms of the classical special fu
nctions for special values of the parameters. Thes
e solutions of the Painlev\\'e equations are often
called ``classical solutions" and frequently can
be expressed in the form of determinants.In the ge
neric case of P$_{\\rm V}$ when $\\delta\\not=0$\,
special function solutions are expressed in terms
of Kummer functions and has rational solutions ex
pressed in terms of Laguerre polynomials. In the c
ase of P$_{\\rm V}$ when $\\delta=0$\, which is kn
own as deg-P$_{\\rm V}$ and related to the third P
ainlev\\'e equation\, special function solutions a
re expressed in terms of Bessel functions and has
algebraic solutions expressed in terms of Laguerre
polynomials. I shall give some new representation
s of some of these classical solutions and discuss
some applications. \;
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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