A theta function identity of degree eight and Eisenstein series identities

Liu, Zhiguo

doi:10.1016/j.jnt.2012.06.007

Cited by 8 publications

(8 citation statements)

References 7 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These two theorems not only allow us to recover several well-known results in elliptic modular functions, but also lead to several new results in [46]. However, many other applications of these two theorems are not discussed.…”

Section: Andmentioning

confidence: 80%

See 1 more Smart Citation

A universal identity for theta functions of degree eight and applications

Liu

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Andmentioning

confidence: 80%

“…We [46,Theorem 1.2] proved the following two remarkable identities for theta functions of degree eight.…”

Section: Andmentioning

confidence: 99%

A universal identity for theta functions of degree eight and applications

Liu

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…These two theorems not only allow us to recover several well-known results in elliptic modular functions, but also lead to several new results in [Li12]. However, many other applications of these two theorems are not discussed.…”

Section: Introduction and Preliminarymentioning

confidence: 80%

A universal identity for theta functions of degree eight and applications

Liu

2021

Hardy-Ramanujan Journal

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Making use of the method of asymptotic analysis (see [5,6]), we can deduce the following trigonometric sums identity.…”

Section: (32)mentioning

confidence: 99%