Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral

Abstract

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite’s two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitz determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painlevé equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

Appendix A: Verification of (5.4)

Usually, we often normalize the Fuchsian system (5.1) so that $A_{N+2}=-\sum _{i=0}^{N+1} A_i$ (the residue matrix at $z=\infty$) is diagonal when considering its isomonodromic deformation; see e.g. [10]. But, in this paper, we adopt a different normalization treating the two points $z=0$ and $z=\infty$ equally; i.e. we choose $A_{N+1}$ (the residue matrix at $z=0$) and $A_{N+2}$ to be upper and lower triangular, respectively. Note that the latter normalization is more versatile in a general setting than the former. In this appendix, for a supplement to Sect. 5.1, we demonstrate how to determine the coefficient $B_i=B_i(z)$ of the deformation equation (5.2):

$$\begin{aligned} \frac{\partial Y}{\partial u_i} =B_i Y \end{aligned}$$

of (5.1).

Differentiating (5.3) with respect to $u_j$ tells us that

• if $i \ne j$ then $\displaystyle \frac{\partial Y}{\partial u_j} Y^{-1}$ is holomorphic at $z=u_i$;

• if $i=j$ then $\displaystyle \frac{\partial Y}{\partial u_i} Y^{-1}= - \frac{A_i}{z-u_i} +(\text {holomorphic at } z=u_i)$.

Similarly, we observe that

$$\begin{aligned} \frac{\partial Y}{\partial u_i} Y^{-1}=\frac{\partial \Psi }{\partial u_i} \Psi ^{-1} = \begin{bmatrix} 0&\quad *&\quad \cdots&\quad * \\&\quad 0&\quad \ddots&\quad \vdots \\&~&\quad \ddots&\quad * \\&~&~&\quad 0 \end{bmatrix} +O(z) \quad \text {near } z=0 \end{aligned}$$

(A.1)

and

$$\begin{aligned} \frac{\partial Y}{\partial u_i} Y^{-1}=\frac{\partial \Phi }{\partial u_i} \Phi ^{-1} = \begin{bmatrix} *&~&\\ \vdots&\quad \ddots&\\ *&\quad \cdots&\quad * \end{bmatrix} +O(w) \quad \text {near } z=1/w=\infty . \end{aligned}$$

(A.2)

Here we have used the assumption that the connection matrix C does not depend on $u_i$. Consequently, $\displaystyle B_i= \frac{\partial Y}{\partial u_i} Y^{-1}$ is a rational function matrix in z that has only a simple pole at $z=u_i$ with residue $-A_i$ and thus

$$\begin{aligned} B_i= \frac{A_i}{u_i-z}+K_i, \end{aligned}$$

where $K_i$ is a constant matrix. Substituting $z=\infty$ ($w=0$) in (A.2) shows that $K_i$ is a lower triangular matrix. Therefore, substituting $z=0$ in (A.1), we conclude that $K_i=- (A_i)_\mathrm{LT}/u_i$.