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.

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.

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 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.

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

FOURTEEN OCTONARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 37-50 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Sum Of Divisors ◽  
Recent Evaluation

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

scholarly journals CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals Representations by octonary quadratic forms with coefficients 1, 2, 3 or 6

2017 ◽  
Vol 13 (03) ◽  
pp. 735-749 ◽  
Author(s):  
Ayşe Alaca ◽  
M. Nesibe Kesicioğlu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms

Using modular forms, we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

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