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TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF $n$ BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS

Part of: Other special functions Forms and linear algebraic groups Diophantine equations Series expansions

Published online by Cambridge University Press:  08 January 2020

ZHI-HONG SUN Open the ORCID record for ZHI-HONG SUN [Opens in a new window]
ZHI-HONG SUN*
Affiliation:
School of Mathematics and Statistics,Huaiyin Normal University, Huaian, Jiangsu 223300, PR China email zhsun@hytc.edu.cn
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Abstract

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 $t(1,1,6,8;n)$.

Keywords

theta functiontriangular numberquadratic form

MSC classification

Primary: 11D85: Representation problems
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E25: Sums of squares and representations by other particular quadratic forms 30B10: Power series (including lacunary series) 33E20: Other functions defined by series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 39 - 49
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the National Natural Science Foundation of China (Grant No. 11771173).

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