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.

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.

TWO THETA-FUNCTION IDENTITIES OF LEVEL 10

2020 ◽  
Vol 9 (7) ◽  
pp. 4929-4936
Author(s):  
D. Anu Radha ◽  
B. R. Srivatsa Kumar ◽  
S. Udupa
Keyword(s):  
Theta Function ◽  
Theta Function Identities

TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS

2020 ◽  
Vol 102 (1) ◽  
pp. 39-49
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Theta Function ◽  
Linear Combinations ◽  
Triangular Numbers ◽  
Transformation Formulas ◽  
Positive Integers ◽  
Theta Function Identities

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

New truncated theorems for three classical theta function identities

2022 ◽  
Vol 101 ◽  
pp. 103470
Author(s):  
Ernest X.W. Xia ◽  
Ae Ja Yee ◽  
Xiang Zhao
Keyword(s):  
Theta Function ◽  
Theta Function Identities

A Pair of Theta Function Identities (L. Carlitz)

SIAM Review ◽  
1974 ◽  
Vol 16 (4) ◽  
pp. 553-555
Author(s):  
G. E. Andrews
Keyword(s):  
Theta Function ◽  
Theta Function Identities

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?.

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