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1: 31.2 Differential Equations
§31.2(v) Heun’s Equation Automorphisms
F -Homotopic Transformations
Homographic Transformations
Composite Transformations
Except for the identity automorphism, each alters the parameters.
2: 31.8 Solutions via Quadratures
Lastly, λ j , j = 1 , 2 , , 2 g + 1 , are the zeros of the Wronskian of w + ( m ; λ ; z ) and w - ( m ; λ ; z ) . By automorphisms from §31.2(v), similar solutions also exist for m 0 , m 1 , m 2 , m 3 , and Ψ g , N ( λ , z ) may become a rational function in z . …
3: 31.3 Basic Solutions
§31.3(iii) Equivalent Expressions
Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). There are 192 automorphisms in all, so there are 192 / 8 = 24 equivalent expressions for each of the 8. …
4: Bibliography T
  • F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
  • 5: Bibliography S
  • C. L. Siegel (1971) Topics in Complex Function Theory. Vol. II: Automorphic Functions and Abelian Integrals. Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience [John Wiley & Sons Inc.], New York.