scholarly journals Some Ramanujan-type circular summation formulas

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ji-Ke Ge ◽  
Qiu-Ming Luo
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Summation Formulas ◽  
Transformation Formulas ◽  
Theta Function Identities ◽  
The Way

AbstractIn this paper, we give two Ramanujan-type circular summation formulas by applying the way of elliptic functions and the properties of theta functions. As applications, we obtain the corresponding imaginary transformation formulas for Ramanujan-type circular summations and some 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 $t(1,1,6,8;n)$.

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.

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.

On the number of representations of integers by sums of mixed numbers

2016 ◽  
Vol 12 (04) ◽  
pp. 945-954
Author(s):  
Ernest X. W. Xia ◽  
Y. H. Ma ◽  
L. X. Tian
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Theta Functions ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Representations Of Integers ◽  
Theta Function Identities

In this paper, several explicit formulas for the number of representations of a positive integer by sums of mixed numbers are determined by employing theta function identities and the [Formula: see text]-parametrization of theta functions due to Alaca, Alaca and Williams. It is interesting that the formulas proved in this paper are linear combinations of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

ON THE REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2012 ◽  
Vol 09 (01) ◽  
pp. 189-204 ◽  
Author(s):  
ERNEST X. W. XIA ◽  
OLIVIA X. M. YAO
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Quadratic Forms ◽  
Theta Functions ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Theta Function Identities

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit formulas are obtained for the number of representations of a positive integer n by the quadratic forms [Formula: see text] with a ≠ 0, a + b + c = 4 and [Formula: see text] with k + l = 2 and r + s + t = 2 by employing these identities.

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 Some Theta-Function Identities Related to Jacobi’s Triple-Product Identity

2018 ◽  
Vol 11 (1) ◽  
pp. 1 ◽  
Author(s):  
Hari M. Srivastava ◽  
M. P. Chaudhary ◽  
Sangeeta Chaudhary
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

The main object of this paper is to present some q-identities involving some of the theta functions of Jacobi and Ramanujan. These q-identities reveal certain relationships among three of the theta-type functions which arise from the celebrated Jacobi’s triple-product identity in a remarkably simple way. The results presented in this paper are motivated by some recent works by Chaudhary et al. (see [4] and [5]) and others (see, for example, [1] and [13]).

scholarly journals A universal identity for theta functions of degree eight and applications

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Theta Function ◽  
Natural Extension ◽  
Theta Functions ◽  
Common Source ◽  
Modular Functions ◽  
Sigma Function ◽  
Function Identity ◽  
International Audience ◽  
Theta Function Identity ◽  
Theta Function Identities

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.

THE NUMBER OF REPRESENTATIONS OF AN INTEGER AS A SUM INVOLVING GENERALIZED PENTAGONAL NUMBERS

2012 ◽  
Vol 08 (04) ◽  
pp. 1041-1056
Author(s):  
OLIVIA X. M. YAO ◽  
ERNEST X. W. XIA
Keyword(s):  
Natural Number ◽  
Theta Function ◽  
Theta Functions ◽  
Quadratic Polynomials ◽  
Theta Function Identities

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. From these identities, we obtain some formulas for the number of representations of a natural number as a sum of quadratic polynomials involving generalized pentagonal numbers. In particular, we derive a formula for the number of representations of a natural number as a sum of twelve generalized pentagonal numbers.

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