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On reducing the Heun equation to the hypergeometric equation
 J. Differential Equations
, 2005
"... The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heuntohypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve ..."
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The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heuntohypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its crossratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higherdegree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. [See K. Kuiken, “Heun’s equation and the hypergeometric equation”, SIAM Journal on Mathematical Analysis 10 (3)
Transforming the Heun equation to the hypergeometric equation, I: Polynomial transformations
 SIAM J. MATH. ANAL
"... The reductions of the Heun equation to the hypergeometric equation by rational changes of its independent variable are classified. Heuntohypergeometric transformations are analogous to the classical hypergeometric identities (i.e., hypergeometrictohypergeometric transformations) of Goursat. How ..."
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The reductions of the Heun equation to the hypergeometric equation by rational changes of its independent variable are classified. Heuntohypergeometric transformations are analogous to the classical hypergeometric identities (i.e., hypergeometrictohypergeometric transformations) of Goursat. However, a transformation is possible only if the singular point location parameter and normalized accessory parameter of the Heun equation are each restricted to take values in a discrete set. The possible changes of variable are all polynomial. They include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higherdegree transformations.
Algebraic solutions of the Lamé equation, revisited
 J. Differential Equations
"... A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, “On algebraic solutions of Lamé’s differential equation”, J. Differential Equatio ..."
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Cited by 4 (1 self)
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A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, “On algebraic solutions of Lamé’s differential equation”, J. Differential Equations 41 (1) (1981), 44–58.] It is shown that if the group is the octahedral group ¡£ ¢ , then the degree parameter of the equation may differ by ¤¦¥¨§� © from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation. [See R. C. Churchill, “Twogenerator subgroups of ������������ � and the hypergeometric, Riemann, and Lamé equations”, J. Symbolic Computation 28 (4–5) (1999), 521–545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.