Addition formulas for Jacobi theta functions, Dedekind's eta function, and Ramanujan's congruences

• Autores: Zhi-Guo Liu
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 240, Nº 1, 2009, págs. 135-150
• Idioma: inglés
• DOI: 10.2140/pjm.2009.240.135
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• Resumen

  • Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities. Here, we use this addition formula to derive a curious theta function identity, which includes Jacobi�s quartic identity and some other important theta function identities as special cases. We give new series expansions for ?2(t), ?6(t), ?8(t), and ?10(t), where ?(t) is Dedekind�s eta function. The series expansions for ?6(t) and ?10(t) lead to simple proofs of Ramanujan�s congruences p(7n + 5) = 0 (mod 7) and p(11n + 6) = 0 (mod 11), respectively.