<p>We introduce a sequence of orthogonal polynomials whose associated moments are the Rayleigh-type sums, involving the zeros of the Bessel function derivative <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2489_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$J_\nu '$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>J</mi> <mi>ν</mi> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> of order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2489_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>. We also discuss the fundamental properties of those polynomials such as recurrence, orthogonality, etc. Consequently, we obtain a formula for the Hankel determinant, elements of which are chosen as the aforementioned Rayleigh-type sums. As an application, we complete the Hurwitz-type theorem for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2489_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$J_\nu '$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>J</mi> <mi>ν</mi> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation>, which deals with the number of complex zeros of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2489_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$J_\nu '$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>J</mi> <mi>ν</mi> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> depending on the range of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2489_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>.</p>

Complex Zeros of Bessel Function Derivatives and Associated Orthogonal Polynomials

期刊论文
发表日期：

2025年8月1日

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Seok-Young Chung,
Sujin Lee,
Young Woong Park

摘要

We introduce a sequence of orthogonal polynomials whose associated moments are the Rayleigh-type sums, involving the zeros of the Bessel function derivative $J_\nu '$ $J_{ν}^{'}$ of order $\nu $ $ν$ . We also discuss the fundamental properties of those polynomials such as recurrence, orthogonality, etc. Consequently, we obtain a formula for the Hankel determinant, elements of which are chosen as the aforementioned Rayleigh-type sums. As an application, we complete the Hurwitz-type theorem for $J_\nu '$ $J_{ν}^{'}$ , which deals with the number of complex zeros of $J_\nu '$ $J_{ν}^{'}$ depending on the range of $\nu $ $ν$ .