For any positive integers t and n, let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1335_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">${\overline{D}}_t(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of t-colored overpartitions of n, where each part in the partition has t distinct colors. In recent years, several authors established many infinite families of congruences for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1335_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">${\overline{D}}_t(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by considering different values of t. In this paper, we prove some infinite and particular congruences for the t-colored overpartition function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1335_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">${\overline{D}}_t(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1335_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="438" /> </InlineMediaObject> <EquationSource Format="TEX">$t = 2; 8m + 2; 8m + 5; 8m + 6; 16m + 2; 16m + 4\,\,\textrm{and}\,\,32m + 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mn>8</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>;</mo> <mn>8</mn> <mi>m</mi> <mo>+</mo> <mn>5</mn> <mo>;</mo> <mn>8</mn> <mi>m</mi> <mo>+</mo> <mn>6</mn> <mo>;</mo> <mn>16</mn> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mo>;</mo> <mn>16</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mtext>and</mtext> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mn>32</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, where m is any non-negative integer.

Some new congruences for t-colored overpartition function

期刊论文
发表日期：

2025年5月30日

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Pujashree Buragohain,
Nipen Saikia

摘要

For any positive integers t and n, let ${\overline{D}}_t(n)$ ${\bar{D}}_{t} (n)$ denote the number of t-colored overpartitions of n, where each part in the partition has t distinct colors. In recent years, several authors established many infinite families of congruences for ${\overline{D}}_t(n)$ ${\bar{D}}_{t} (n)$ by considering different values of t. In this paper, we prove some infinite and particular congruences for the t-colored overpartition function ${\overline{D}}_t(n)$ ${\bar{D}}_{t} (n)$ for $t = 2; 8m + 2; 8m + 5; 8m + 6; 16m + 2; 16m + 4\,\,\textrm{and}\,\,32m + 4$ $t = 2; 8 m + 2; 8 m + 5; 8 m + 6; 16 m + 2; 16 m + 4 and 32 m + 4$ , where m is any non-negative integer.