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On the holonomic deformation of linear differential equations of \(A_4\) type. (English) Zbl 0908.34006
This paper deals with holonomic deformations of a linear differential equation of the form \[ \begin{aligned} &\frac{d^2y}{dx^2}+p_1(x,t)\frac{dy}{dx}+p_2(x,t)y=0, \tag{1}\\ &p_1(x,t)=-2x^5-\sum_{j=1}^4jt_jx^{j-1}-\sum_{k=1}^4\frac 1{x-\lambda_k}, \\ &p_2(x,t)=-(2\alpha +1)x^4-2\sum_{j=1}^4H_jx^{4-j}+\sum_{k=1}^4 \frac{\mu_k}{x-\lambda_k},\end{aligned} \] with \((2\alpha +1\not\in\mathbb{Z}),\) where none of the four regular singularities \(x=\lambda_k\) \((k=1,\dots,4)\) is a logarithmic singularity. By a nonlogarithmic condition, explicit forms of \(H_i\) \((i=1,\dots,4)\) are determined to be rational functions of \(t=(t_1,\dots,t_4), \lambda= (\lambda_1,\dots,\lambda_4)\) and \(\mu=(\mu_1,\dots,\mu_4).\) The main result says that the holonomic deformation of (1) is governed by a completely integrable Hamiltonian system of the form \[ \frac{\partial\lambda_k}{\partial t_j}=\frac{\partial \overline H_j} {\partial\mu_k},\quad \frac{\partial\mu_k}{\partial t_j}=-\frac {\partial\overline H_j}{\partial\lambda_k}\quad (k,j=1,\dots,4). \] Here, each Hamiltonian function \(\overline H_j\) is a linear function of \((H_1,\dots,H_4)\) whose coefficients are polynomials in \((t_1,\dots,t_4)\).
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A30 Linear ordinary differential equations and systems
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