On the schröter formula for theta functions

Zhi Guo Liu; Xiao Mei Yang

doi:10.1142/S1793042109002754

On the schröter formula for theta functions

Zhi Guo Liu^*, Xiao Mei Yang

^*Corresponding author for this work

School of Mathematical Sciences

East China Normal University

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

Original language	English
Pages (from-to)	1477-1488
Number of pages	12
Journal	International Journal of Number Theory
Volume	5
Issue number	8
DOIs	https://doi.org/10.1142/S1793042109002754
State	Published - Dec 2009

Keywords

Addition formula
Elliptic function
Modular equation
Rogers-Ramanujan's continued fraction
Schröter formula
Septuple product identity
Theta function

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10.1142/S1793042109002754

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title = "On the schr{\"o}ter formula for theta functions",

abstract = "The Schr{\"o}ter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schr{\"o}ter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.",

keywords = "Addition formula, Elliptic function, Modular equation, Rogers-Ramanujan's continued fraction, Schr{\"o}ter formula, Septuple product identity, Theta function",

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year = "2009",

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AU - Liu, Zhi Guo

AU - Yang, Xiao Mei

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N2 - The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

AB - The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

KW - Addition formula

KW - Elliptic function

KW - Modular equation

KW - Rogers-Ramanujan's continued fraction

KW - Schröter formula

KW - Septuple product identity

KW - Theta function

UR - https://www.scopus.com/pages/publications/77149164432

U2 - 10.1142/S1793042109002754

DO - 10.1142/S1793042109002754

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SN - 1793-0421

VL - 5

SP - 1477

EP - 1488

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 8

ER -

On the schröter formula for theta functions

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Keywords

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