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 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].

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 Number of Representations of Integers by Quadratic Forms in Twelve Variables

1998 ◽  
Vol 5 (6) ◽  
pp. 545-564
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Positive Integers

Abstract A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.

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 ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

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 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.

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