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<p>In this paper, we give a new proof of Hirose–Sato’s formula for the expansion of <Equation ID="Equ4"> <EquationSource Format="TEX">\( \zeta (\{2\}^{a_1-1},3,\dots ,\{2\}^{a_r-1},3,\{2\}^{c-1},1,\{2\}^{b_1},\dots , 1,\{2\}^{b_s}) \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <mrow> <msub> <mi>a</mi> <mi>r</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>3</mn> <mo>,</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>c</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mo>,</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> </msup> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>1</mn> <mo>,</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <msub> <mi>b</mi> <mi>s</mi> </msub> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </Equation>in the Hoffman basis, using the drop 1 relation.</p>

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Diamond lift of Hirose–Sato’s formula involving the Hoffman basis

  • 期刊论文

  • 发表日期：

    2026年7月10日

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  • Shin-ichiro Seki

摘要

In this paper, we give a new proof of Hirose–Sato’s formula for the expansion of \( \zeta (\{2\}^{a_1-1},3,\dots ,\{2\}^{a_r-1},3,\{2\}^{c-1},1,\{2\}^{b_1},\dots , 1,\{2\}^{b_s}) \) ζ ( { 2 } a 1 - 1 , 3 , ⋯ , { 2 } a r - 1 , 3 , { 2 } c - 1 , 1 , { 2 } b 1 , ⋯ , 1 , { 2 } b s ) in the Hoffman basis, using the drop 1 relation.