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<p>In this paper, we establish the generating differential formula for the weakly modular forms of weight 16 and 20, using the family of functions depending on the Eisenstein series of weight 2<i>k</i>, for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_661_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le k\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, and denoted by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_661_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(E_{2k}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. These differential formulas are of assistance in the process of finding solutions to a variety of nonlinear differential equations. In addition to this, we show that these formulas are related to the integer partitions of 8 and 10.</p>

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Differential formula for weakly modular forms of weight 16 and 20

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Ambreen Ahmed,
  • Aykut Ahmet Aygunes,
  • M. P. Chaudhary

摘要

In this paper, we establish the generating differential formula for the weakly modular forms of weight 16 and 20, using the family of functions depending on the Eisenstein series of weight 2k, for \(1\le k\in \mathbb {N}\) 1 ≤ k ∈ N , and denoted by \(E_{2k}\) E 2 k . These differential formulas are of assistance in the process of finding solutions to a variety of nonlinear differential equations. In addition to this, we show that these formulas are related to the integer partitions of 8 and 10.