Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_{\ell ,k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> represent the number of k-tuple <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-regular partitions of n. In this study, we investigate the arithmetic behavior of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_{5^{2k-1},6}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>6</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. To analyze this, we first derive congruences for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$T_{5,6}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mn>5</mn> <mo>,</mo> <mn>6</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo powers of 5, then construct generating functions for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_{5^{2k-1},6}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <msup> <mn>5</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>6</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> along certain arithmetic progressions. Through this approach, we derive infinite families of Ramanujan-type congruences that hold modulo powers of 5.

On congruences modulo powers of 5 for k-tuple $\ell $-regular partitions

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2026年4月24日

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Gouri Shankar Guru,
Yudhisthira Jamudulia

摘要

Let $T_{\ell ,k}(n)$ $T_{ℓ, k} (n)$ represent the number of k-tuple $\ell $ $ℓ$ -regular partitions of n. In this study, we investigate the arithmetic behavior of $T_{5^{2k-1},6}(n)$ $T_{5^{2 k - 1}, 6} (n)$ . To analyze this, we first derive congruences for $T_{5,6}(n)$ $T_{5, 6} (n)$ modulo powers of 5, then construct generating functions for $T_{5^{2k-1},6}(n)$ $T_{5^{2 k - 1}, 6} (n)$ along certain arithmetic progressions. Through this approach, we derive infinite families of Ramanujan-type congruences that hold modulo powers of 5.

On congruences modulo powers of 5 for k-tuple \(\ell \)-regular partitions

摘要