Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_{k,\ell }(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> count the number of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(k,\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-regular bipartitions of n. In this paper, we establish infinite families of congruences modulo powers of 5 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_{5^{2k-1}, 5^{2k}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, for any integers <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$k \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that <Equation ID="Equ34"> <EquationSource Format="TEX">$\begin{aligned} B_{5^{2k-1}, 5^{2k}} \left( 5^{2k+2\beta -1} n + \dfrac{2 \cdot 5^{2k+\beta } - 3 \cdot 5^{2k-1} + 1}{12} \right) \equiv 0 \pmod {5^{k+\beta }}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>B</mi> <mrow> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </msub> <mfenced close=")" open="("> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>n</mi> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mo>·</mo> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mi>β</mi> </mrow> </msup> <mo>-</mo> <mn>3</mn> <mo>·</mo> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>12</mn> </mfrac> </mstyle> </mfenced> <mo>≡</mo> <mn>0</mn> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <msup> <mn>5</mn> <mrow> <mi>k</mi> <mo>+</mo> <mi>β</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>by deriving the exact generating functions of specific arithmetic progressions in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$B_{5^{2k-1}, 5^{2k}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This result substantially extends the earlier findings of Tang (Quaestiones Mathematicae 2020, 43(2): 169-183).

Generalized congruences for $(k,\ell )$-regular bipartitions modulo powers of 5

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发表日期：

2026年1月14日

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C. Shivashankar,
Nithin Anthony Kumar,
N. Girisha

摘要

Let $B_{k,\ell }(n)$ $B_{k, ℓ} (n)$ count the number of $(k,\ell )$ $(k, ℓ)$ -regular bipartitions of n. In this paper, we establish infinite families of congruences modulo powers of 5 for $B_{5^{2k-1}, 5^{2k}}(n)$ $B_{5^{2 k - 1}, 5^{2 k}} (n)$ , for $k \ge 1$ $k \geq 1$ . In particular, for any integers $n \ge 0$ $n \geq 0$ , $\beta \ge 0$ $β \geq 0$ and $k \ge 1$ $k \geq 1$ , we prove that $\begin{aligned} B_{5^{2k-1}, 5^{2k}} \left( 5^{2k+2\beta -1} n + \dfrac{2 \cdot 5^{2k+\beta } - 3 \cdot 5^{2k-1} + 1}{12} \right) \equiv 0 \pmod {5^{k+\beta }}, \end{aligned}$ $\begin{matrix} B_{5^{2 k - 1}, 5^{2 k}} (5^{2 k + 2 β - 1} n + \frac{2 \cdot 5^{2 k + β} - 3 \cdot 5^{2 k - 1} + 1}{12}) \equiv 0 (mod 5^{k + β}), \end{matrix}$ by deriving the exact generating functions of specific arithmetic progressions in $B_{5^{2k-1}, 5^{2k}}(n)$ $B_{5^{2 k - 1}, 5^{2 k}} (n)$ . This result substantially extends the earlier findings of Tang (Quaestiones Mathematicae 2020, 43(2): 169-183).

Generalized congruences for \((k,\ell )\)-regular bipartitions modulo powers of 5

摘要