scholarly journals The explicit formulas and evaluations of Ramanujan's theta-function ψ

2006 ◽  
Vol 321 (1) ◽  
pp. 157-181 ◽  
Author(s):  
Jinhee Yi ◽  
Yang Lee ◽  
Dae Hyun Paek
Keyword(s):  
Theta Function ◽  
Explicit Formulas ◽  
Ramanujan’S Theta Function

ON THE QUATERNARY FORMS x2+y2+2z2+3t2, x2+2y2+2z2+6t2, x2+3y2+3z2+6t2 AND 2x2+3y2+6z2+6t2

2012 ◽  
Vol 08 (07) ◽  
pp. 1661-1686 ◽  
Author(s):  
AYŞE ALACA ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Quadratic Forms ◽  
Ramanujan’S Theta Function

Formulas are proved for the number of representations of a positive integer by each of the four quaternary quadratic forms x2+y2+2z2+3t2, x2+2y2+2z2+6t2, x2+3y2+3z2+6t2 and 2x2+3y2+6z2+6t2. As a consequence of these formulas, each of the four series [Formula: see text] is determined in terms of Ramanujan's theta function.

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.

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].

Ramanujan’s theta function identities and the relations between sums of squares and sums of triangular numbers

2020 ◽  
Vol 16 (06) ◽  
pp. 1275-1294
Author(s):  
Min Bian ◽  
Shane Chern ◽  
Doris D. M. Sang ◽  
Ernest X. W. Xia
Keyword(s):  
Theta Function ◽  
Nonnegative Integer ◽  
Sums Of Squares ◽  
Triangular Numbers ◽  
Positive Integers ◽  
Ramanujan’S Theta Function ◽  
Theta Function Identities

For positive integers [Formula: see text], [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of representations of a nonnegative integer [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative integers, and let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are integers. Recently, Sun proved a number of relations between [Formula: see text] and [Formula: see text] along with numerous conjectures on such relations. In this work, we confirm several conjectures of Sun by using Ramanujan’s theta function identities.

scholarly journals Some Congruence Properties of a Restricted Bipartition Function cN(n)

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Nipen Saikia ◽  
Chayanika Boruah
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Ramanujan’S Theta Function ◽  
Congruence Properties ◽  
Theta Function Identities

Let cN(n) denote the number of bipartitions (λ,μ) of a positive integer n subject to the restriction that each part of μ is divisible by N. In this paper, we prove some congruence properties of the function cN(n) for N=7, 11, and 5l, for any integer l≥1, by employing Ramanujan’s theta-function identities.

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 Explicit Values for Ramanujan's Theta Function ϕ(q)

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Bruce C Berndt ◽  
Örs Rebák
Keyword(s):  
Theta Function ◽  
Rational Number ◽  
Theta Functions ◽  
Positive Rational Number ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook ◽  
Ramanujan’S Theta Function ◽  
First Descriptions

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

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