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<p>On p. &#xa0;206 in his lost notebook, Ramanujan recorded an incomplete septic theta function identity. Motivated by the completion of this identity by the second author, we offer cubic and quintic analogues. Using the theory generated by these two analogues and Ramanujan’s class invariants, we provide many evaluations for Ramanujan’s most prominent theta function, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1013_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi (q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in his notation.</p>

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Cubic and quintic analogues of Ramanujan’s septic theta function identity

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月12日

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  • Bruce C. Berndt,
  • Örs Rebák

摘要

On p.  206 in his lost notebook, Ramanujan recorded an incomplete septic theta function identity. Motivated by the completion of this identity by the second author, we offer cubic and quintic analogues. Using the theory generated by these two analogues and Ramanujan’s class invariants, we provide many evaluations for Ramanujan’s most prominent theta function, \(\varphi (q)\) φ ( q ) in his notation.