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<p>In this present paper, we show that the Stirling numbers of the first kind with higher level are connected with the probability distribution of the number of records and record times in the so-called <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1004_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(F^{\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation>-scheme. In addition, we provide an alternative proof of the formula for determining the location of the maximum of these numbers by using the distribution of the number of records. Moreover, we establish an explicit formula for the maximum of these numbers. As an application, some examples are provided at the end.</p>

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Stirling numbers with higher level and records

  • 期刊论文

  • 发表日期：

    2025年1月29日

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  • F. A. Shiha

摘要

In this present paper, we show that the Stirling numbers of the first kind with higher level are connected with the probability distribution of the number of records and record times in the so-called \(F^{\alpha }\) F α -scheme. In addition, we provide an alternative proof of the formula for determining the location of the maximum of these numbers by using the distribution of the number of records. Moreover, we establish an explicit formula for the maximum of these numbers. As an application, some examples are provided at the end.