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Sums of Squares and Sums of Triangular Numbers
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Shaun Cooper
Published/Copyright:
March 10, 2010
Abstract
Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form 
, where 位1 + . . . + 位饾憵 = 2, 4 or 8. We also give a new and simple combinatorial proof of the modular equation of order seven.
Received: 2006-10-20
Published Online: 2010-03-10
Published in Print: 2006-December
漏 Heldermann Verlag
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Articles in the same Issue
- Finitely Generated Modules and Connectivity
- Triangulated Track Categories
- Principal Maps in Triangulated Categories and Maps between 2-Stage Spectra
- A Local System of Spaces and a Bigraded Model of Fibration
- On the Lacunarity of Two-Eta-Products
- Sums of Squares and Sums of Triangular Numbers
- New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms
- Quadratic Envelope of the Category of Class Two Nilpotent Groups
- On the Basis of the Space of Cusp Forms of the Level 12
- On the Eisenstein Series Corresponding to Quadratic Forms of Certain Type
- Completing the Operadic Butterfly
- Representation of Numbers by Quadratic Forms. Main Results of the Research Done in Georgia
- Sums of Odd Squares
- Some Properties of the Inversion Counting Function
- On Cusp Forms of Weight 9/2
- On a Double Series of Chan and Ong
- Coprimeness among Element Orders of Finite Groups
Keywords for this article
Sum of squares;
sum of triangular numbers;
theta function;
modular equation
Articles in the same Issue
- Finitely Generated Modules and Connectivity
- Triangulated Track Categories
- Principal Maps in Triangulated Categories and Maps between 2-Stage Spectra
- A Local System of Spaces and a Bigraded Model of Fibration
- On the Lacunarity of Two-Eta-Products
- Sums of Squares and Sums of Triangular Numbers
- New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms
- Quadratic Envelope of the Category of Class Two Nilpotent Groups
- On the Basis of the Space of Cusp Forms of the Level 12
- On the Eisenstein Series Corresponding to Quadratic Forms of Certain Type
- Completing the Operadic Butterfly
- Representation of Numbers by Quadratic Forms. Main Results of the Research Done in Georgia
- Sums of Odd Squares
- Some Properties of the Inversion Counting Function
- On Cusp Forms of Weight 9/2
- On a Double Series of Chan and Ong
- Coprimeness among Element Orders of Finite Groups
