Some Infinite Products of Ramanujan Type

Alaca, Ayşe ; Alaca, Şaban ; Williams, Kenneth S.

Canadian mathematical bulletin, Tome 52 (2009) no. 4, pp. 481-492

Résumé

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.

Détail

DOI : 10.4153/CMB-2009-050-5

Mots-clés : 11E25, 11F11, 11F27, 30B10, Power series expansions of certain infinite products

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