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1
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Some new infinite families of congruences modulo 3 for overpartitions into odd parts

100%
Xia E.
Colloquium Mathematicum
|
2016
|
tom 142
|
nr 2
255-266
EN
Let $p̅_{o}(n)$ denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $p̅_{o}(n)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $p̅_{o}(n)$ modulo 3. For example, we prove that for n, α ≥ 0, $p̅_{o}(4^{α}(24n+17)) ≡ p̅_{o}(4^{α}(24n+23)) ≡ 0 (mod 3)$.
2
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New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

100%
Xia E.
Colloquium Mathematicum
|
2015
|
tom 140
|
nr 1
91-105
EN
Let $\overline{pp}(n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\overline{pp}(3n+2) ≡ 0 (mod 3)$. They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\overline{pp}(n)$. Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\overline{pp}(n)$. Furthermore, they also constructed infinite families of congruences for $\overline{pp}(n)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite families of congruences modulo 9 for $\overline{pp}(n)$. For example, we find that for all integers k,n ≥ 0, $\overline{pp}(2^{6k}(48n+20)) ≡ \overline{pp}(2^{6k}(384n+32)) ≡ \overline{pp}(2^{3k}(48n+36)) ≡ 0 (mod 9)$.
3
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On the number of representations of a positive integer by certain quadratic forms

100%
Xia E.
Colloquium Mathematicum
|
2014
|
tom 135
|
nr 1
139-145
EN
For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form $∑_{i=1}^{a} (x²_i + x_iy_i + y²_i) + 2∑_{j=1}^{b} (u²_j + u_jv_j + v²_j)$. Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.
4
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Proof of a conjecture of Hirschhorn and Sellers on overpartitions

64%
Chen W., Xia E.
Acta Arithmetica
|
2014
|
tom 163
|
nr 1
59-69
EN
Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod 8) for n ≥ 0 obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we confirm the conjecture of Hirschhorn and Sellers.
5
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Representation numbers of five sextenary quadratic forms

51%
Xia E., Yao O., Zhao A.
Colloquium Mathematicum
|
2015
|
tom 138
|
nr 2
247-254
EN
For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form $∑_{i=1}^{a} (x²_i + x_iy_i + y²_i) + 2∑_{j=1}^{b} (u²_j + u_jv_j + v²_j) + 4∑_{k=1}^{c} (r²_k + r_ks_k + s²_k)$. Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.
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