Middle convolution and deformation for Fuchsian systems

Abstract

Middle convolution and addition are operations for Fuchsian systems of differential equations which preserve the number of accessory parameters. In this paper we show that they also preserve the deformation equations. Several Bäcklund transformations of the sixth Painlevé equation are obtained from this viewpoint.

Prolongability of Ordinary Differential Equations

“…It is a fundamental operation for Fuchsian systems, which leaves the number of accessory parameters and the deformation equation invariant ( [7], [1], [5]). We like to consider a relation between the middle convolution and prolongability.…”

“…We have shown that these operations leaves the deformation equation invariant [5]. Then we are interested in the relation between the prolongation and these operations.…”

A Higher Order Painlevé System in Two Variables and Extensions of the Appell Hypergeometric Functions <i>F</i>₁, <i>F</i>₂ and <i>F</i>₃

“…Since the Fuchsian system of type 11 Â 5 is transformed by Katz's two operations, namely addition and middle convolution, to the one of type 21 Â 5, the Painlevé system H 11Â5 is equivalent to H 21Â5 ; see [7]. It is the same for the other systems.…”

“…Thanks to the previous works [7,10,12,13,14] we have a good classification theory of isomonodromy deformation equations of Fuchsian systems; we call them higher order Painlevé systems. From this point of view several extensions of the Painlevé VI equation have been proposed in [5,6,11,15,19,21].…”

Katz's middle convolution and Yokoyama's extending operation

“…Hence, the property of Katz's operation can be transferred to Yokoyama's operations and vice versa. For example, it is proved by [5] that the middle convolution preserves the deformation equation and therefore so do Yokoyama's operations.…”