scholarly journals New modular equations of signature three in the spirit of Ramanujan

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2847-2868
Author(s):  
Kumar Srivatsa ◽  
S Shruthi
Keyword(s):  
Theta Function ◽  
Continued Fractions ◽  
Theta Functions ◽  
Modular Equations ◽  
Class Invariants ◽  
Theta Function Identities

Srinivasa Ramanujan recorded many modular equations in his notebooks, which are useful in the computation of class invariants, continued fractions and the values of theta functions. In this paper, we prove some new modular equations of signature three by using theta function identities of composite degrees.

scholarly journals An alternate circular summation formula of theta functions and its applications

2012 ◽  
Vol 6 (1) ◽  
pp. 114-125 ◽  
Author(s):  
Jun-Ming Zhu
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Summation Formula ◽  
Theta Function Identities

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for (q; q)2n?.

SOME INVERSE RELATIONS AND THETA FUNCTION IDENTITIES

2012 ◽  
Vol 08 (08) ◽  
pp. 1977-2002 ◽  
Author(s):  
ZHI-GUO LIU
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Theta Function ◽  
Theta Functions ◽  
Fourier Series Expansion ◽  
Theta Function Identities ◽  
Trigonometric Identities

Two pairs of inverse relations for elliptic theta functions are established with the method of Fourier series expansion, which allow us to recover many classical results in theta functions. Many nontrivial new theta function identities are discovered. Some curious trigonometric identities are derived.

ON THE SCHRÖTER FORMULA FOR THETA FUNCTIONS

2009 ◽  
Vol 05 (08) ◽  
pp. 1477-1488 ◽  
Author(s):  
ZHI-GUO LIU ◽  
XIAO-MEI YANG
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Modular Equations ◽  
Trigonometric Identity ◽  
Product Identity ◽  
Function Identity ◽  
Special Cases ◽  
Theta Function Identity ◽  
Addition Formulas

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.

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

2008 ◽  
Vol 04 (03) ◽  
pp. 461-474 ◽  
Author(s):  
PEE CHOON TOH
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Root Systems ◽  
Jacobi Theta Functions ◽  
Multiple Variables ◽  
Affine Root Systems ◽  
Theta Function Identities

We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.

scholarly journals Some New Explicit Values of Quotients of Theta-Function ϕ(q) and Applications to Ramanujan's Continued Fractions

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Nipen Saikia
Keyword(s):  
Theta Function ◽  
Continued Fractions ◽  
Modular Equations ◽  
Real Numbers ◽  
Positive Real ◽  
Ramanujan’S Theta Function

We find some new explicit values of the parameter hk,n for positive real numbers k and n involving Ramanujan's theta-function ϕ(q) and give some applications of these new values for the explicit evaluations of Ramanujan's continued fractions. In the process, we also establish two new identities for ϕ(q) by using modular equations.

scholarly journals Some Somos’s theta function identities of level 6 and application to partitions

2020 ◽  
Vol 108 (122) ◽  
pp. 137-144
Author(s):  
Belakavadi Radhakrishna Srivatsa Kumar ◽  
Gururaj Sharath
Keyword(s):  
Theta Function ◽  
Modular Equations ◽  
Programmable Calculator ◽  
Colored Partitions ◽  
Combinatorial Interpretations ◽  
Theta Function Identities

M. Somos discovered around 6200 theta function identities using PARI/GP scripts without offering the proof. He runs PARI/GP scripts and it works as a sophisticated programmable calculator. These identities highly resemble those of Ramanujan?s identities. Here we prove a few theta-function identities of level 6 discovered by Somos by using modular equations of degree 3 given by Ramanujan and further we extract some interesting combinatorial interpretations of colored partitions.

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