On the two-dimensional theta functions of the Borweins

Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S.

doi:10.4064/aa124-2-4

Cited by 45 publications

(32 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Throughout this paper q denotes a complex variable such that |q| < 1. In this paper we determine in a uniform manner a formula for N (a, b, c, d; n), valid for all n ∈ N, for each of the nineteen quaternary quadratic forms given by (a, b, c, d) = (1, 1, 1, 1), (1,1,1,4), (1,1,2,2), (1,1,3,3), (1,1,3,12), (1.3) (1, 1, 4, 4), (1,1,6,6), (1,1,12,12), (1, 2, 2, 4), (1,2,3,6), (1,3,3,4), (1,3,4,12), (1,4,4,4), (1,4,6,6), (1,4,12,12), (2,2,3,…”

mentioning

confidence: 99%

Nineteen quaternary quadratic forms

Alaca

Lemire

et al. 2007

Acta Arith.

Self Cite

View full text Add to dashboard Cite

mentioning

confidence: 99%

Nineteen quaternary quadratic forms

Alaca

Lemire

et al. 2007

Acta Arith.

Self Cite

View full text Add to dashboard Cite

“…For a, b, c, d ∈ N and n ∈ N ∪ {0}, we define N (a, b, c, d; n) = card{(x, y, z, t) ∈ Z 4 | n = ax 2 + by 2 + cz 2 + dt 2 }. There are ten quaternary quadratic forms ax 2 +by 2 +cz 2 +dt 2 with a, b, c, d ∈ {1, 3, 9} and 1 = a ≤ b ≤ c ≤ d. Formulae for N (a, b, c, d; n) (n ∈ N) with (a, b, c, d) = (1, 1, 1, 1), (1,1,1,3), (1,1,3,3), (1,3,3,3) appear in the literature (see [1], [2]). In this paper we treat the remaining six forms (1, 1, 1, 9), (1,1,3,9), (1,1,9,9), (1,3,3,9), (1,3,9,9) and (1,9,9,9).…”

mentioning

confidence: 99%

“…228, 231] to determine explicit formulae for N (1, 1, 3, 9; n) and N (1, 3, 9, 9; n) in terms of A(n), B(n), C(n) and D(n) (see Theorems 1.2 and 1.3). Finally, we use the (p, k)-parametrization of ϕ(q) given in [3] and [5] to determine explicit formulae for N (1, 1, 1, 9; n), N (1, 1, 9, 9; n) and N (1, 9, 9, 9; n) in terms of σ(n/k) (k ∈ {1, 2, 3, 6, 9, 36}) and the integers c(n) (n ∈ N) defined by…”

mentioning

confidence: 99%

See 1 more Smart Citation

Representations by quaternary quadratic forms whose coefficients are 1, 3 and 9

Alaca

2009

Acta Arith.

Self Cite

View full text Add to dashboard Cite

“…Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams [1], they showed that…”

Section: Introductionmentioning

confidence: 99%

Ramanujan-type congruences for overpartitions modulo 16

et al. 2015

View full text Add to dashboard Cite

Abstract. Let p(n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that p(4n + 3) ≡ 0 (mod 8) for n ≥ 0. They also conjectured that p(40n + 35) ≡ 0 (mod 40) for n ≥ 0. Chen and Xia proved this conjecture by using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that p(5n) ≡ (−1) n p(4 · 5n) (mod 5) for n ≥ 0 and p(n) ≡ (−1) n p(4n) (mod 8) for n ≥ 0 by using the relation of the generating function of p(5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p(n) due to Mahlburg. As a consequence, we deduce that p(4 k (40n + 35)) ≡ 0 (mod 40) for n, k ≥ 0. Furthermore, applying the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform, we obtain an infinite family of congrences p(4 k · 5ℓ 2 n) ≡ 0 (mod 5), where k ≥ 0 and ℓ is a prime such that ℓ ≡ 3 (mod 5) and −n ℓ = −1. Moreover, we show that p(5 2 n) ≡ p(5 4 n) (mod 5) for n ≥ 0. So we are led to the congruences p 4 k 5 2i+3 (5n ± 1) ≡ 0 (mod 5) for n, k, i ≥ 0. In this way, we obtain various Ramanujan-type congruences for p(n) modulo 5 such as p(45(3n + 1)) ≡ 0 (mod 5) and p(125(5n ± 1)) ≡ 0 (mod 5) for n ≥ 0.

show abstract

On the two-dimensional theta functions of the Borweins

Cited by 45 publications

References 4 publications

Nineteen quaternary quadratic forms

Nineteen quaternary quadratic forms

Representations by quaternary quadratic forms whose coefficients are 1, 3 and 9

Ramanujan-type congruences for overpartitions modulo 16

Contact Info

Product

Resources

About