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TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF
$n$ BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 1 / August 2020
- Published online by Cambridge University Press:
- 08 January 2020, pp. 39-49
- Print publication:
- August 2020
-
- Article
- Export citation
-
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 
$t(1,1,6,8;n)$.
