We prove arithmetic properties of some restricted overpartition functions in which the parts are from certain residue classes of 8. For example, if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{p}_{3,4,5}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{p}_{1,4,7}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of overpartitions of a positive integer n into parts congruent to 3, 4, or 5 modulo 8 and congruent to 1, 4, or 7 modulo 8, respectively, then <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{p}_{3,4,5}(16n+1)\equiv 0~(\text {mod}~16)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="3.33333pt" /> <mn>16</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\overline{p}_{1,4,7}(16n+9)\equiv 0~(\text {mod}~16)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="3.33333pt" /> <mn>16</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all non-negative integers n.

On some restricted overpartition functions

期刊论文
发表日期：

2025年3月12日

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Nayandeep Deka Baruah,
Subhajit Bandyopadhyay

摘要

We prove arithmetic properties of some restricted overpartition functions in which the parts are from certain residue classes of 8. For example, if $\overline{p}_{3,4,5}(n)$ ${\bar{p}}_{3, 4, 5} (n)$ and $\overline{p}_{1,4,7}(n)$ ${\bar{p}}_{1, 4, 7} (n)$ denote the number of overpartitions of a positive integer n into parts congruent to 3, 4, or 5 modulo 8 and congruent to 1, 4, or 7 modulo 8, respectively, then $\overline{p}_{3,4,5}(16n+1)\equiv 0~(\text {mod}~16)$ ${\bar{p}}_{3, 4, 5} (16 n + 1) \equiv 0 (mod 16)$ and $\overline{p}_{1,4,7}(16n+9)\equiv 0~(\text {mod}~16)$ ${\bar{p}}_{1, 4, 7} (16 n + 9) \equiv 0 (mod 16)$ for all non-negative integers n.