Discrete integrable systems generated by Hermite-Padé approximants (Q2811972)

The authors study the connection between discrete integrable systems on \(\mathbb Z^2\) satisfying the \textit{discrete zero curvature condition} represented by a field of square invertible matrices NEWLINE\[NEWLINEL_{n,m},M_{n,m}\in\mathbb C^{d\times d},\quad n,m\in\mathbb Z,NEWLINE\]NEWLINE with the conditon NEWLINE\[NEWLINEL_{n,m+1}M_{n,m}-M_{n+1,m}L_{n,m}=0,NEWLINE\]NEWLINE (forming a \textit{Lax pair} on \(\mathbb Z^2\)) on one hand and rational Hermite-Padé approximants on the other hand. These are given for a pair \(\vec{f}=(f_1,f_2)\) of Laurent series at infinity NEWLINE\[NEWLINEf_j(z)=\sum_{k=0}^{\infty}\quad{s_{j,k}\over z^{k+1}},\quad j=1,2,NEWLINE\]NEWLINE by the type II pair NEWLINE\[NEWLINE\pi_{\vec{n}}=\left({Q_{\vec{n}}^{(1)}\over P_{\vec{n}}},{Q_{\vec{n}}^{(2)}\over P_{\vec{n}}}\right)NEWLINE\]NEWLINE for the multi-index \(\vec{n}=(n_1,n_2)\in\mathbb N^2\), defined by NEWLINE\[NEWLINE\deg P_{\vec{n}}\leq |\vec{n}|=n_1+n_2,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf_j(z)P_{\vec{n}}(z)-Q_{\vec{n}}^{(j)}={\mathcal O}\left({1\over z^{n_j+1}}\right),\;z\rightarrow\infty.NEWLINE\]NEWLINE If all the multi-indices are \textit{normal} (i.e. the rational approximant is unique and has full degree \(\deg P_{n_1,n_2}=n_1+n_2\)), the pair \(\vec{f}\) is called \textit{perfect}. In this case the transition matrices of the recurrence relations expressing \(P_{n+1,m}\) and \(P_{n,m+1}\) in terms of \(P_{n,m},P_{n-1,m},P_{n,m-1}\) are \(3\times 3\) matrices of the special form NEWLINE\[NEWLINEL_{n,m}=\left(\begin{matrix}{x+\alpha_{n,m}^{(1)}} & \alpha_{n,m}^{(2)} & \alpha_{n,m}^{(3)} \cr \alpha_{n,m}^{(4)} & 0 & 0 \cr \alpha_{n,m}^{(5)} & 0 & 1\cr\end{matrix}\right), M_{n,m}=\left(\begin{matrix}{x+\beta_{n,m}^{(1)}} & \alpha_{n,m}^{(2)} & \alpha_{n,m}^{(3)} \cr \alpha_{n,m}^{(4)} & 1 & 0 \cr \alpha_{n,m}^{(5)} & 0 & 0\cr\end{matrix}\right).NEWLINE\]NEWLINENEWLINENEWLINEThe main result is now given inNEWLINENEWLINETheorem 1.1. The zero curvature condition holds for a family of \(3\times 3\) transition matrices \(L_{n,m}\) and \(M_{n,m}\) of the form given above, if and only if there exists a perfect system of two functions such that the \(P_{n,m}\) are the Hermite-Padé polynomials with recurrence coefficents of the form indicated in these transition matrices.NEWLINENEWLINEMoreover, they give in Theorem 1.1 a nonlinear system of difference equations to calculate the coefficients of the recurrence relation, i.e. the entries of the matrices forming the Lax pair.NEWLINENEWLINEThe paper is divided into five sections and a list of references (35 items):NEWLINENEWLINE1. IntroductionNEWLINENEWLINE2. The generic Lax representationNEWLINENEWLINE3. Orthogonal polynomials via \(2\times 2\) matrix polynomialsNEWLINENEWLINE4. Hermite-Padé and multiple orthogonal polynomialsNEWLINENEWLINE5. The underlying boundary value problemNEWLINENEWLINEHere two types of perfect systems are indicated: Angelesco systems, Nikishin systems.