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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Hypergeometric solutions to the symmetric discrete
Painlev equations - Kajiwara\, K (Kyushu)
DTSTART;TZID=Europe/London:20090703T153000
DTEND;TZID=Europe/London:20090703T163000
UID:TALK18973AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/18973
DESCRIPTION:The discrete Painlev'e equations are usually expre
ssed in the form of system of first-order ordinary
difference equations\, but it is possible to redu
ce them to single second-order difference equation
s by imposing certain conditions on parameters. Th
e former generic equations are sometimes called ``
asymmetric''\, and latter ``symmetric''\, referrin
g to the terminology of the QRT mapping. A typical
example is a discrete Painlev'e II equation (dP$_
{\nm II}$) egin{displaymath} x_{n+1}+x_{n-1} = r
ac{(an+b)x_n+c}{1-x_n^2}\, nd{displaymath} and th
e ``asymmetric'' discrete Painlev'e II equation (a
dP$_{\nm II}$) egin{displaymath} Y_{n+1} + Y_{n}
= rac{(2a n+b)X_n + c+d}{1-X_n^2}\,quad X_{n+1} +
X_{n} = rac{(a(2n+1)+b)Y_{n+1} + c-d}{1-Y_{n+1}^
2}. nd{displaymath} dP$_{\nm II}$ is derived by i
mposing the constraint $d=0$ on adP$_{\nm II}$ and
putting $X_n=x_{2n}$\, $Y_n=x_{2n-1}$\, respectiv
ely. \nadP$_{\nm II}$ arises as the B"acklund tran
sformation of P$_{\nm V}$\, and hence its hypergeo
metric solutions are expressed by the Hankel deter
minant whose entries are given by the confluent hy
pergeometric functions. However\, the above specia
lization does not yield the hypergeometric solutio
ns to dP$_{\nm II}$ which are given by egin{displ
aymath} x_n=rac{2}{z}~rac{ au_{N+1}^{n+1} au_N^n
}{ au_{N+1}^n au_N^{n+1}}-1\,quad au_N^n=detleft(
H_{n+2i+j-3}\night)_{i\,j=1\,ldots\,N}. nd{displa
ymath} Here $H_n$ is the parabolic cylinder functi
on satisfying egin{displaymath} H_{n+1}-zH_n + nH
_{n-1}=0\, nd{displaymath} and $a=rac{8}{z^2}$\,
$b=rac{4(1+2N)}{z^2}$ and $c=-rac{4(1+2N)}{z^2}
$ ($Ninmathbb{Z}_{geq 0}$). More precisely\, (i) t
he asymmetric structure of shifts in the determina
nt\, and (ii) the entry $H_n$\, cannot be recovere
d by putting $d=0$ in the hypergeometric solutions
to adP$_{\nm II}$. Such ``inconsistency'' among t
he hypergeometric solutions to symmetric and asymm
etric discrete Painlev'e equations has been observ
ed already in the first half of 90's\, but left un
solved for a long time. Moreover\, the determinant
with similar asymmetric shift cannot be seen for
the solutions to other integrable systems. \n\nIn
this talk\, we consider the $q$-Painlev'e equation
of type $widetilde{W}(A_2+A_1)^{(1)}$ ($q$-P$_{\n
m III}$) as an example\, and clarify the mechanism
of the above phenomena by using the birational re
presentation of the Weyl group. \n\nThis work has
been done in collaboration with N. Nakazono and T.
Tsuda (Kyushu Univ.).\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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