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# Some Infinite Products of Ramanujan Type

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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 - 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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