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Some Infinite Products of Ramanujan Type

  Published:2009-12-01
 Printed: Dec 2009
  • Ay\c{s}e Alaca
  • \c{S}aban Alaca
  • Kenneth S. Williams
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Abstract

In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities \[ \prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})} =1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n \] and \[ q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})} =\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n. \] We give several more identities of this type.
Keywords: Power series expansions of certain infinite products Power series expansions of certain infinite products
MSC Classifications: 11E25, 11F11, 11F27, 30B10 show english descriptions Sums of squares and representations by other particular quadratic forms
Holomorphic modular forms of integral weight
Theta series; Weil representation; theta correspondences
Power series (including lacunary series)
11E25 - Sums of squares and representations by other particular quadratic forms
11F11 - Holomorphic modular forms of integral weight
11F27 - Theta series; Weil representation; theta correspondences
30B10 - Power series (including lacunary series)
 

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