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<p>Recently, Wang gave a systematic study on the parity of coefficients of classical mock theta functions. Very recently, Chen and Garvan proved some congruences modulo 4 for five of Ramanujan’s mock theta functions. Kaur and Rana studied congruences for Ramanujan’s sixth-order mock theta function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1018_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho (q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the arithmetic progressions 2<i>n</i>. In this paper, we prove some new congruences modulo 4 and 8 for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1018_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho (q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the arithmetic progressions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1018_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(2n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> by using some results on the Hurwitz class number due to Chen and Garvan and an identity proved by Mortenson.</p>

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Congruences modulo 4 and 8 for Ramanujan’s sixth-order mock theta function \(\rho (q)\)

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Yueya Hu,
  • Eric H. Liu,
  • Olivia X. M. Yao

摘要

Recently, Wang gave a systematic study on the parity of coefficients of classical mock theta functions. Very recently, Chen and Garvan proved some congruences modulo 4 for five of Ramanujan’s mock theta functions. Kaur and Rana studied congruences for Ramanujan’s sixth-order mock theta function \(\rho (q)\) ρ ( q ) in the arithmetic progressions 2n. In this paper, we prove some new congruences modulo 4 and 8 for \(\rho (q)\) ρ ( q ) in the arithmetic progressions \(2n+1\) 2 n + 1 by using some results on the Hurwitz class number due to Chen and Garvan and an identity proved by Mortenson.