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The Ramanujan Journal, ISSN 1382-4090, 4/2019, Volume 48, Issue 3, pp. 639 - 654
Let T(a, b, c, d; n) denote the number of representations of n as $$a\frac{x(x+1)}{2}+ b\frac{y(y+1)}{2}+c\frac{z(z+1) }{2}+d\frac{w(w+1)}{2}$$ a x ( x + 1 ) 2... 
11E25 | Fourier Analysis | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | 11D85 | Mathematics | Theta function identities | Number Theory | Combinatorics | Sum of triangular numbers | MATHEMATICS
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 5/2018, Volume 46, Issue 1, pp. 1 - 18
...Ramanujan J (2018) 46:1–18 https://doi.org/10.1007/s11139-016-9841-x Congruences modulo 64 and 1024 for overpartitions Olivia X. M. Yao 1 Received: 29 May 2016... 
Overpartitions | 11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | RAMANUJAN-TYPE CONGRUENCES | POWERS | SQUARES
Journal Article
Nature Genetics, ISSN 1061-4036, 01/2017, Volume 49, Issue 1, pp. 125 - 130
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 10/2018, Volume 47, Issue 1, pp. 221 - 235
.... Xia 1 · Olivia X. M. Yao 1 Received: 12 January 2017 / Accepted: 13 February 2017 / Published online: 29 April 2017 © Springer Science+Business Media New York... 
11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | t -Core partitions | Mathematics | Number Theory | Combinatorics | 05A17 | t-Core partitions | MATHEMATICS | CORE PARTITIONS | FACTS
Journal Article
Graphs and Combinatorics, ISSN 0911-0119, 1/2015, Volume 31, Issue 1, pp. 299 - 307
...Graphs and Combinatorics (2015) 31:299–307 DOI 10.1007/s00373-013-1378-9 ORIGINAL PAPER Two Formulas of 2-Color Off-Diagonal Rado Numbers Olivia X. M. Yao... 
Off-diagonal Rado number | 05D10 | 05C55 | Schur number | 11B75 | Mathematics | Engineering Design | Combinatorics | Rado number | MATHEMATICS | A(X+Y)=BZ | Graphs | Integers | Texts | Mathematical analysis | Combinatorial analysis | Color
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 8/2017, Volume 43, Issue 3, pp. 619 - 631
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 05/2014, Volume 10, Issue 3, pp. 669 - 674
Let bt(n) denote the number of t-regular partitions of n. Recently, Keith discovered some congruences modulo 3 for 9-regular partitions. He also conjectured... 
Regular partitions | congruences | MATHEMATICS
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 5/2014, Volume 34, Issue 1, pp. 109 - 117
...Ramanujan J (2014) 34:109–117 DOI 10.1007/s11139-013-9493-z Parity results for 9-regular partitions Ernest X.W. Xia · Olivia X.M. Yao Received: 23 October 2012... 
Partition | 11P83 | Fourier Analysis | Congruence | Functions of a Complex Variable | Field Theory and Polynomials | Regular partition | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | IDENTITIES | PARTS | DISTINCT
Journal Article
Journal of applied mathematics, ISSN 1110-757X, 9/2012, Volume 2012, pp. 1 - 14
.... In their paper, J. M. Borwein and P. B. Borwein [1] introduced the following three functions: (1.1)a(q)=∑m,n=-∞∞qm2+mn+n2,b(q)=∑m,n=-∞∞ωm-nqm2+mn+n2,c(q)=∑m,n=-∞∞q(m+1/3)2... 
RAMANUJAN | MATHEMATICS, APPLIED | MODULAR IDENTITIES | CONVOLUTION SUMS | Series | Theorems (Mathematics) | Functions | Research | Functional equations | Studies | Mathematical models | Convolution | Elliptic functions | Mathematical analysis | Sums
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 8/2013, Volume 31, Issue 3, pp. 373 - 396
... $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues... 
Partitions | 11P83 | Fourier Analysis | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | Partition congruences | Mathematics | Number Theory | Combinatorics | 2-Dissections | 05A17 | MATHEMATICS | PROOFS | BEAUTIFUL IDENTITY | EQUATIONS | MODULAR RELATIONS | GOLLNITZ-GORDON FUNCTIONS | Heterocyclic compounds
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 06/2015, Volume 11, Issue 4, pp. 1063 - 1072
The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let k(n) denote the number of Fu's k dots bracelet... 
partitions | k dots bracelet partitions | Congruence | MATHEMATICS | CONGRUENCES MODULO POWERS | Jewelry
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 02/2014, Volume 10, Issue 1, pp. 115 - 123
In this paper, the convolution sums ∑i+3j=nσ(i)σ3(j) and ∑3i+j=nσ(i)σ3(j) are evaluated for all n ∈ ℕ, which answers an open problem given by Huard, Ou,... 
sum of divisors function | Eisenstein series | convolution sums
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 09/2014, Volume 10, Issue 6, pp. 1421 - 1430
In this paper, the convolution sum ∑i+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams. 
Journal Article
Nature (London), ISSN 0028-0836, 11/2018, Volume 563, Issue 7729, pp. 72 - 78
Journal Article
Indian Journal of Pure and Applied Mathematics, ISSN 0019-5588, 2/2014, Volume 45, Issue 1, pp. 53 - 74
Journal Article
Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, ISSN 0308-2105, 10/2019, Volume 149, Issue 5, pp. 1189 - 1205
.... For example, we prove that a(((15 X 17(3 alpha) +1)/8)) equivalent to (mod 5) for alpha > 0. 
cubic partitions | congruences | θ function identities | theta function identities | MATHEMATICS | MATHEMATICS, APPLIED | CONTINUED-FRACTION | RAMANUJANS | Partitions (mathematics) | Partitions | Congruences
Journal Article
Ramanujan Journal, ISSN 1382-4090, 05/2017, Volume 43, Issue 1, pp. 215 - 228
In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions,... 
Singular overpartitions | Theta functions | Arithmetic properties | MATHEMATICS | ANALOGS
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 1/2013, Volume 30, Issue 1, pp. 1 - 7
...Ramanujan J (2013) 30:1–7 DOI 10.1007/s11139-012-9385-7 On the 4-dissections of certain infinite products Olivia X.M. Yao · Ernest X.W. Xia Received: 30... 
Fourier Analysis | Ramanujan’s continued fraction | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | 11A55 | Mathematics | Number Theory | Infinite product | 30B70 | Combinatorics | Ramanujan's continued fraction | MATHEMATICS | RAMANUJANS CONTINUED-FRACTION | Surgery
Journal Article
NATURE, ISSN 0028-0836, 09/2019, Volume 573, Issue 7772, pp. 61 - 61
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 02/2013, Volume 9, Issue 1, pp. 189 - 204
In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit... 
quadratic form | Eisenstein series | sum of squares | MATHEMATICS | IDENTITIES | SUMS
Journal Article
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