THETA FUNCTION IDENTITIES AND REPRESENTATIONS BY CERTAIN QUATERNARY QUADRATIC FORMS

2008 ◽  
Vol 04 (02) ◽  
pp. 219-239 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
MATHIEU F. LEMIRE ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Quadratic Forms ◽  
Theta Function Identities

Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms.

THE NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY CERTAIN QUATERNARY QUADRATIC FORMS

2009 ◽  
Vol 05 (01) ◽  
pp. 13-40 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
MATHIEU F. LEMIRE ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Quadratic Forms ◽  
Theta Function Identities

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x2 + ey2 + fz2 + gt2 with e, f, g ∈ {1, 2, 4, 8}.

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.

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.

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].

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.

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
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