Generating Function Associated with the Hankel Determinant Formula for the Solutions of the Painleve IV Equation

Abstract

Abstract. We consider a Hankel determinant formula for generic solutions of the Painlevé IV equation. We show that the generating functions for the entries of the Hankel determinants are related to the asymptotic solution at infinity of the isomonodromic problem. Summability of these generating functions is also discussed.

A remark on the Hankel determinant formula for solutions of the Toda equation

“…In other words, the relations between determinant formula for the solution of P II and auxiliary linear problems originate from the structure of the Toda equation. We also note that one can recover the results for the Painlevé IV equation [3,11] in similar manner.…”

Toda chain, Stieltjes function, and orthogonal polynomials

“…It has several advantages for investigating connections to the Painlevé equations (see, e.g., [10]). In this section, we consider this formalism from the standpoint of orthogonal polynomials.…”

Pade Method to Painleve Equations

“…Recently, the connection between the Hankel determinant expression of the t-function of the Toda equations and the auxiliary linear problems has been clarified [6], and the similar phenomena observed in Painlevé equations [4] [5] were recovered from this. The above argument may give another explanation of these results from the point of view of iso-monodromy deformation.…”

“…These results have been obtained from the Riccati solutions (see for example [2] [8]), via computational method using the Bäcklund or Schlesinger transformations [10] [11] [12] [13] [14]. An interesting explanation of these determinant structures (not only for the special solutions but also for generic cases) has recently given in [4] [5] [6] from the point of view of the Toda equations.…”