On poles of the rational solution of the Toda equation of Painlevé - II type
The Toda equation q'\({}_ n=p_{n-1}-p_ n\), p'\({}_ n=p_ n(q_ n- q_{n+1})\), \(n=0,\pm 1,\pm 2,..\). admits the special rational solution \(q_ n=(\log P_ n/P_{n+1})'\), \(p_ n=(\log P_{n+1})''-t/4\) where \(p_ n=\prod^{d(n)}_{k=1}(t-a_{n,k})\) with \(d(n)=n(n-1)/2\). All zeros of \(P_ n\) are simple, \(P_ n\) and \(P_{n+1}\) have no common zero. So \(q_ n\) has \(n^ 2\) simple poles and \(p_ n\) has \(n(n+1)/2\) double poles. A sharp estimate for the maximal modulus of these poles is obtained \[ (n(n+3))^{1/3}\leq\max \{| a_{n+2,k}|;1\leq k\leq d(n+2)\}\leq (2(11+5\sqrt{5}))^{1/3}n^{2/3},\quad n=1,2,3,... . \]
scientific article; zbMATH DE number 3869782