<p>Ramanujan recorded seventeen mock theta functions in his last letter to Hardy. Quite recently, Das (J Math Anal Appl 543(2):128913, 2025) proved some congruences modulo small powers of 3 between the coefficients of the second order mock theta functions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu _2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$A_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, introduced by Ramanujan and McIntosh, respectively. Moreover, Das conjectured a congruence modulo 2187 and three congruences modulo 6561 between the coefficients of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu _2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$A_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove these conjectural congruences by employing some <i>q</i>-series manipulations. Further, we conjecture an infinite family of congruences modulo high powers of 3 between the coefficients of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu _2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1728_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$A_2(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Proof of a conjecture of Das on the coefficients of mock theta functions

期刊论文
发表日期：

2025年4月15日

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Dazhao Tang

摘要

Ramanujan recorded seventeen mock theta functions in his last letter to Hardy. Quite recently, Das (J Math Anal Appl 543(2):128913, 2025) proved some congruences modulo small powers of 3 between the coefficients of the second order mock theta functions $\mu _2(q)$ $μ_{2} (q)$ and $A_2(q)$ $A_{2} (q)$ , introduced by Ramanujan and McIntosh, respectively. Moreover, Das conjectured a congruence modulo 2187 and three congruences modulo 6561 between the coefficients of $\mu _2(q)$ $μ_{2} (q)$ and $A_2(q)$ $A_{2} (q)$ . In this paper, we prove these conjectural congruences by employing some q-series manipulations. Further, we conjecture an infinite family of congruences modulo high powers of 3 between the coefficients of $\mu _2(q)$ $μ_{2} (q)$ and $A_2(q)$ $A_{2} (q)$ .