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On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia — 2014

Colloquium Mathematicae

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 ( x ² i + x i y i + y ² i ) + 2 j = 1 b ( u ² j + u j v j + v ² 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.

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

Ernest X. W. Xia — 2016

Colloquium Mathematicae

Let p ̅ 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 p ̅ 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 p ̅ o ( n ) modulo 3. For example, we prove that for n, α ≥ 0, p ̅ o ( 4 α ( 24 n + 17 ) ) p ̅ o ( 4 α ( 24 n + 23 ) ) 0 ( m o d 3 ) .

New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia — 2015

Colloquium Mathematicae

Let p p ¯ ( n ) denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, p p ¯ ( 3 n + 2 ) 0 ( m o d 3 ) . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for p p ¯ ( n ) . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for p p ¯ ( n ) . Furthermore, they also constructed infinite families of congruences for p p ¯ ( n ) modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several new infinite...

Proof of a conjecture of Hirschhorn and Sellers on overpartitions

William Y. C. ChenErnest X. W. Xia — 2014

Acta Arithmetica

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...

Representation numbers of five sextenary quadratic forms

Ernest X. W. XiaOlivia X. M. YaoA. F. Y. Zhao — 2015

Colloquium Mathematicae

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 ( x ² i + x i y i + y ² i ) + 2 j = 1 b ( u ² j + u j v j + v ² j ) + 4 k = 1 c ( r ² k + r k s k + s ² 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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