<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{p}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of overpartitions of <i>n</i>. In this paper, we establish the generating function for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{p}(96n+12)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mn>96</mn> <mi>n</mi> <mo>+</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 81 using elementary dissection techniques and theta function identities. Based on this generating function and some identities of Ramanujan theta functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi (q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi (-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we obtain a congruence relation and an infinite family of congruences modulo 81 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\overline{p}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, by studying the periodicity of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi (q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> based on the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-dissection formula of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\psi (q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> given by Cui and Gu, we find some arithmetic relations and infinite families of congruences modulo 3 and 27 for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\overline{p}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Arithmetic Relations Modulo 3, 27 and 81 for Overpartitions

期刊论文
发表日期：

2025年2月14日

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Liang Jin,
Li Zhang

摘要

Let $\overline{p}(n)$ $\bar{p} (n)$ denote the number of overpartitions of n. In this paper, we establish the generating function for $\overline{p}(96n+12)$ $\bar{p} (96 n + 12)$ modulo 81 using elementary dissection techniques and theta function identities. Based on this generating function and some identities of Ramanujan theta functions $\psi (q)$ $ψ (q)$ and $\varphi (-q)$ $φ (- q)$ , we obtain a congruence relation and an infinite family of congruences modulo 81 for $\overline{p}(n)$ $\bar{p} (n)$ . Furthermore, by studying the periodicity of $\psi (q)$ $ψ (q)$ based on the $\ell $ $ℓ$ -dissection formula of $\psi (q)$ $ψ (q)$ given by Cui and Gu, we find some arithmetic relations and infinite families of congruences modulo 3 and 27 for $\overline{p}(n)$ $\bar{p} (n)$ .