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Some Infinite Products of Ramanujan Type |
Canad. Math. Bull. 52(2009), 481-492
Published:2009-12-01
Printed: Dec 2009
Printed: Dec 2009
- Ay\c{s}e Alaca
- \c{S}aban Alaca
- Kenneth S. Williams
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 |
| 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) |