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On representations by figurate numbers: a uniform approach to the conjectures of Melham. (English) Zbl 1281.11033

The famous two-square theorem of Jacobi can be expressed in terms of Lambert series as \[ \sum_{x,y=-\infty}^{\infty}q^{x^2+y^2}=1+4\sum_{n=1}^{\infty}\left(\frac{-4}{n}\right)\frac{q^n}{1-q^n}. \] R. S. Melham [Integers 10, No. 1, 83–100 (2010; Zbl 1200.11025)] conjectured 21 analogous formulas with the binary quadratic form \(x^2+y^2\) replaced by various binary sums of triangular, pentagonal and heptagonal numbers. In the present paper, the author gives derivations for these formulas based on formulas for the numbers of representations of positive integers by positive definite binary quadratic forms associated to imaginary quadratic fields of class number 2.

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
11E16 General binary quadratic forms
11D85 Representation problems

Citations:

Zbl 1200.11025
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References:

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