Some Infinite Products of Ramanujan Type
Canadian mathematical bulletin, Tome 52 (2009) no. 4, pp. 481-492
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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.
Mots-clés :
11E25, 11F11, 11F27, 30B10, Power series expansions of certain infinite products
Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. Some Infinite Products of Ramanujan Type. Canadian mathematical bulletin, Tome 52 (2009) no. 4, pp. 481-492. doi: 10.4153/CMB-2009-050-5
@article{10_4153_CMB_2009_050_5,
author = {Alaca, Ay\c{s}e and Alaca, \c{S}aban and Williams, Kenneth S.},
title = {Some {Infinite} {Products} of {Ramanujan} {Type}},
journal = {Canadian mathematical bulletin},
pages = {481--492},
year = {2009},
volume = {52},
number = {4},
doi = {10.4153/CMB-2009-050-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-050-5/}
}
TY - JOUR AU - Alaca, Ayşe AU - Alaca, Şaban AU - Williams, Kenneth S. TI - Some Infinite Products of Ramanujan Type JO - Canadian mathematical bulletin PY - 2009 SP - 481 EP - 492 VL - 52 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-050-5/ DO - 10.4153/CMB-2009-050-5 ID - 10_4153_CMB_2009_050_5 ER -
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