scholarly journals Theta function identities and representation numbers of certain quadratic forms in twelve variables

2017 ◽  
Vol 18 (2) ◽  
pp. 889
Author(s):  
Bülent Köklüce ◽  
Ibrahim Karatay
Keyword(s):  
Theta Function ◽  
Quadratic Forms ◽  
Theta Function Identities

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}.

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.

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.

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?.

scholarly journals Theta function identities from optical neural network transformations

1993 ◽  
Vol 16 (4) ◽  
pp. 805-810
Author(s):  
E. Elizalde ◽  
A. Romeo
Keyword(s):  
Neural Network ◽  
Theta Function ◽  
Heisenberg Model ◽  
Symmetric Functions ◽  
New Approach ◽  
Jacobi Theta Function ◽  
Cylindrically Symmetric ◽  
Optical Neural Network ◽  
Theta Function Identities

We take a new approach to the generation of Jacobi theta function identities. It is complementary to the procedure which makes use of the evaluation of Parseval-like identities for elementary cylindrically-symmetric functions on computer holograms. Our method is more simple and explicit than this one, which was an outcome of the construction of neurocomputer architectures through the Heisenberg model.

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