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<p>We give Euler-like recursive formulas for the <i>t</i>-colored partition function when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t=3,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> as well as for all <i>t</i>-regular partition functions. In particular, we derive an infinite family of “triangular number” recurrences for the 3-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of <i>q</i>-series identities for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((q;q)_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((q;q)_{\infty }^3.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>∞</mi> </mrow> <mn>3</mn> </msubsup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Euler-type recurrences for t-color and t-regular partition functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月9日

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  • Tapas Bhowmik,
  • Wei-Lun Tsai,
  • Dongxi Ye

摘要

We give Euler-like recursive formulas for the t-colored partition function when \(t=2\) t = 2 or \(t=3,\) t = 3 , as well as for all t-regular partition functions. In particular, we derive an infinite family of “triangular number” recurrences for the 3-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of q-series identities for \((q;q)_{\infty }\) ( q ; q ) ∞ and \((q;q)_{\infty }^3.\) ( q ; q ) ∞ 3 .