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

Liu, Zhiguo

doi:10.2140/pjm.2009.240.135

Cited by 18 publications

(8 citation statements)

References 24 publications

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“…Theorem 2.1 is equivalent to [Li07a, Theorem 1], which has been used in [Li07a,Li09] to derive many elliptic function identities, including Ramanujan's cubic theta function identity and Winquist's identity [Wi69]. Next we will give a few applications of Theorem 2.1.…”

Section: A General Theta Function Identity Of Degreementioning

confidence: 99%

A universal identity for theta functions of degree eight and applications

Liu

2021

Hardy-Ramanujan Journal

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International audience Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

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Section: A General Theta Function Identity Of Degreementioning

confidence: 99%

A universal identity for theta functions of degree eight and applications

Liu

2021

Hardy-Ramanujan Journal

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“…Theorem 2.1 is equivalent to [39, Theorem 1], which has been used in [39,42] to derive many elliptic function identities, including Ramanujan's cubic theta function identity and Winquist's identity [65]. Next we will give a few applications of Theorem 2.1.…”

Section: A General Theta Function Identity Of Degreementioning

confidence: 99%

A universal identity for theta functions of degree eight and applications

Liu

2021

Preprint

Self Cite

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Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

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“…When w = 0, the above equation reduces to [17,Theorem 1.3], which includes many well-known addition formulas for the Jacobi theta functions as special cases.…”

Section: A New Addition Formula For Theta Functionsmentioning

confidence: 99%

Kronecker theta function and a decomposition theorem for theta functions I

Liu¹

2020

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The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's 1 ψ 1 summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.Unless otherwise stated, it is assumed throughout that q=exp(2πiτ ), where Imτ > 0.

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

Cited by 18 publications

References 24 publications

A universal identity for theta functions of degree eight and applications

A universal identity for theta functions of degree eight and applications

A universal identity for theta functions of degree eight and applications

Kronecker theta function and a decomposition theorem for theta functions I

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