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Hypergeometric solutions to the symmetric discrete Painlevé equations

Kajiwara, K (Kyushu)
Friday 03 July 2009, 15:30-16:30

Seminar Room 1, Newton Institute


The discrete Painlev\'e equations are usually expressed in the form of system of first-order ordinary difference equations, but it is possible to reduce them to single second-order difference equations by imposing certain conditions on parameters. The former generic equations are sometimes called ``asymmetric'', and latter ``symmetric'', referring to the terminology of the QRT mapping. A typical example is a discrete Painlev\'e II equation (dP$_{\rm II}$) \begin{displaymath} x_{n+1}+x_{n-1} = \frac{(an+b)x_n+c}{1-x_n^2}, \end{displaymath} and the ``asymmetric'' discrete Painlev\'e II equation (adP$_{\rm II}$) \begin{displaymath} Y_{n+1} + Y_{n} = \frac{(2a n+b)X_n + c+d}{1-X_n^2},\quad X_{n+1} + X_{n} = \frac{(a(2n+1)+b)Y_{n+1} + c-d}{1-Y_{n+1}^2}. \end{displaymath} dP$_{\rm II}$ is derived by imposing the constraint $d=0$ on adP$_{\rm II}$ and putting $X_n=x_{2n}$, $Y_n=x_{2n-1}$, respectively. adP$_{\rm II}$ arises as the B\"acklund transformation of P$_{\rm V}$, and hence its hypergeometric solutions are expressed by the Hankel determinant whose entries are given by the confluent hypergeometric functions. However, the above specialization does not yield the hypergeometric solutions to dP$_{\rm II}$ which are given by \begin{displaymath} x_n=\frac{2}{z}~\frac{\tau_{N+1}^{n+1}\tau_N^n}{\tau_{N+1}^n\tau_N^{n+1}}-1,\quad \tau_N^n=\det\left(H_{n+2i+j-3}\right)_{i,j=1,\ldots,N}. \end{displaymath} Here $H_n$ is the parabolic cylinder function satisfying \begin{displaymath} H_{n+1}-zH_n + nH_{n-1}=0, \end{displaymath} and $a=\frac{8}{z^2}$, $b=\frac{4(1+2N)}{z^2}$ and $c=-\frac{4(1+2N)}{z^2}$ ($N\in\mathbb{Z}_{\geq 0}$). More precisely, (i) the asymmetric structure of shifts in the determinant, and (ii) the entry $H_n$, cannot be recovered by putting $d=0$ in the hypergeometric solutions to adP$_{\rm II}$. Such ``inconsistency'' among the hypergeometric solutions to symmetric and asymmetric discrete Painlev\'e equations has been observed already in the first half of 90's, but left unsolved for a long time. Moreover, the determinant with similar asymmetric shift cannot be seen for the solutions to other integrable systems. In this talk, we consider the $q$-Painlev\'e equation of type $\widetilde{W}(A_2+A_1)^{(1)}$ ($q$-P$_{\rm III}$) as an example, and clarify the mechanism of the above phenomena by using the birational representation of the Weyl group. This work has been done in collaboration with N. Nakazono and T. Tsuda (Kyushu Univ.).


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