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Some Infinite Products of Ramanujan Type

Published online by Cambridge University Press:  20 November 2018

Ayşe Alaca ,
Şaban Alaca  and
Kenneth S. Williams
Ayşe Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 e-mail: aalaca@math.carleton.casalaca@math.carleton.cakwilliam@connect.carleton.ca
Şaban Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 e-mail: aalaca@math.carleton.casalaca@math.carleton.cakwilliam@connect.carleton.ca
Kenneth S. Williams
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 e-mail: aalaca@math.carleton.casalaca@math.carleton.cakwilliam@connect.carleton.ca
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Abstract

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In his “lost” notebook, Ramanujan stated two results, which are equivalent to the identities

$$\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{n}} \right)}^{5}}}{\left( 1-{{q}^{5n}} \right)}=1-5\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{d} \right)d \right){{q}^{n}}$$

and

$$q\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{5n}} \right)}^{5}}}{\left( 1-{{q}^{n}} \right)}=\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{n/d} \right)d \right){{q}^{n}}.$$

We give several more identities of this type.

Keywords

11E2511F1111F2730B10Power series expansions of certain infinite products
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 481 - 492
Copyright
Copyright © Canadian Mathematical Society 2009

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