Exact generating functions for the number of partitions into distinct parts

2018 ◽  
Vol 14 (07) ◽  
pp. 1995-2011 ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Nilufar Mana Begum
Keyword(s):  
Theta Function ◽  
Generating Functions ◽  
Continued Fraction ◽  
Negative Integer ◽  
Ramanujan’S Theta Function ◽  
Theta Function Identities

Let [Formula: see text] denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for [Formula: see text], [Formula: see text] and [Formula: see text]. We deduce some congruences modulo 5 and 25. We employ Ramanujan’s theta function identities and some identities for the Rogers–Ramanujan continued fraction.

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.

scholarly journals A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity

2020 ◽  
Vol 108 (122) ◽  
pp. 23-32
Author(s):  
Mahendra Chaudhary
Keyword(s):  
Theta Function ◽  
Open Problem ◽  
Continued Fraction ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

We establish a set of two new relationships involving R?,R? and Rm-functions, which are based on Jacobi?s famous triple-product identity. We, also provide answer for an open problem of Srivastava, Srivastava, Chaudhary and Uddin, which suggest to find an inter-relationships between R?,R? and Rm(m ? N), q-product identities and continued-fraction identities.

Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

2019 ◽  
Vol 15 (01) ◽  
pp. 189-212 ◽  
Author(s):  
Ernest X. W. Xia ◽  
Zhang Yan
Keyword(s):  
Theta Function ◽  
Sums Of Squares ◽  
Triangular Numbers ◽  
Positive Integers ◽  
Ramanujan’S Theta Function ◽  
Theta Function Identities

In a recent paper, Sun posed six conjectures on the relations between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where [Formula: see text] are positive integers, [Formula: see text] are arbitrary nonnegative integers, and [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where this time [Formula: see text] are integers. In this paper, we prove Sun’s six conjectures by using Ramanujan’s theta function identities.

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
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