Application of singular elliptic functions to the investigation of certain properties of Legendre polynomials
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Abstract
In this work, the following theorem is proved. Let m be a negative integer containing no square divisors. Consider the quadratic field k(√m) and let ω1, ω2 be a basis of some ideal of this field, with the imaginary part of the ratio ω2/ω1 being positive. Furthermore, let ρ(u; ω1, ω2, ω3) be the well-known Weierstrass function (in what follows we will index the basis ω1, ω2 such that the ratio ω2/ω1 has a positive imaginary part), which satisfies the following differential equation
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(Archived) Articles
Funding:
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References
- R. Fueter. Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen. Berlin 1924, 1927.
- Tannery et Molk. Éléments de la théorie des fonctions elliptiques 1902. V. IV. p. 91.
- H. Weber. Elliptische Funktionen und algebraische Zahlen. 1908. Table VI.
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