TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS
2020 ◽
Vol 102
(1)
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pp. 39-49
Keyword(s):
Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.
2020 ◽
Vol 16
(06)
◽
pp. 1275-1294
2019 ◽
Vol 15
(01)
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pp. 189-212
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2016 ◽
Vol 12
(04)
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pp. 945-954
Keyword(s):
2020 ◽
Vol 9
(7)
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pp. 4929-4936
2021 ◽
Vol 14
(2)
◽
pp. 380-395
2012 ◽
Vol 6
(1)
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pp. 114-125
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