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Fuchsian differential equations of order 3,...,6 with three singular points and an accessory parameter

Yoshishige Haraoka, Hiroyuki Ochiai, Takeshi Sasaki, Masaaki Yoshida

Abstract

Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with three singular points and one accessory parameter are presented. The shift operators for $H_6$ are studied. They lead to assign the accessory parameter of $H_6$ a cubic polynomial of local exponents so that the equation has several nice symmetries. The other equations will be studied in the forthcoming papers.

Fuchsian differential equations of order 3,...,6 with three singular points and an accessory parameter

Abstract

Fuchsian differential equations of order with three singular points and one accessory parameter are presented. The shift operators for are studied. They lead to assign the accessory parameter of a cubic polynomial of local exponents so that the equation has several nice symmetries. The other equations will be studied in the forthcoming papers.
Paper Structure (82 sections, 50 theorems, 287 equations)

This paper contains 82 sections, 50 theorems, 287 equations.

Table of Contents

  1. Equations $H_j,G_j,E_j\ (j=3,4,5,6)$ and $E_2$
  2. Equation $H_6$
  3. Proof of Proposition \ref{['eqwithR6']}
  4. Table of equations $H_j\ (j=6,5,4,3)$ and $E_2$
  5. Equations $G_j, E_j\ (j=6,5,4,3)$
  6. $G_6(e,a)$
  7. $G_j(e,a)\ (j=3,4,5)$
  8. $E_j(e)\ (j=6,5,4,3)$
  9. Generalities
  10. Symmetry
  11. Shift symmetry
  12. Differentiation, adjoint and coordinate change
  13. Symmetries of $H_j,G_j,E_j$
  14. Examples
  15. $(\theta,\partial)$-form and $(x,\theta,\partial)$-form
  16. ...and 67 more sections

Key Result

Proposition 1.1

The differential equation with the Riemann scheme $R_6$ such that any local solution at 0 and 1 does not have logarithmic terms can be written as eqTxdx with coeffT. This equation has four free coefficients $\{p_{10},p_{20},p_{21},p_{32}\}$. Defining three polynomials $\{f,g,h\}$ by inditialeq and e

Theorems & Definitions (92)

  • Proposition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 82 more