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Ramanujan's theta functions and linear combinations of three triangular numbers | Zhi-Hong Sun
; | Date: |
26 Nov 2018 | Abstract: | Let $Bbb Z$ and $Bbb N$ be the set of integers and the set of positive
integers, respectively. For $a,b,c,ninBbb N$ let $N(a,b,c;n)$ be the number
of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the
number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2 $
$(x,y,zinBbb Z)$. In this paper, by using Ramanujan’s theta functions
$varphi(q)$ and $psi(q)$ we reveal the relation between $t(2,3,3;n)$ and
$N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. In
addition, we pose many conjectures on $t(a,b,c;n)$ for some special values of
$(a,b,c)$. | Source: | arXiv, 1812.3820 | Services: | Forum | Review | PDF | Favorites |
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