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<p>We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the Chebyshev polynomials of both kinds, deriving combinatorial and algebraic properties of some special cases. We show that the zeros of these polynomial sequences lie on certain algebraic curves, some of which we study in greater detail. Finally, we extend part of this work to restricted <i>b</i>-ary overpartitions for arbitrary integers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1187_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(b\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Polynomials and algebraic curves related to certain binary and b-ary overpartitions

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Karl Dilcher,
  • Larry Ericksen

摘要

We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the Chebyshev polynomials of both kinds, deriving combinatorial and algebraic properties of some special cases. We show that the zeros of these polynomial sequences lie on certain algebraic curves, some of which we study in greater detail. Finally, we extend part of this work to restricted b-ary overpartitions for arbitrary integers \(b\ge 2\) b ≥ 2 .