<p>In this work, we derive combinatorial properties of the degenerate falling operators, present their connections with a generalization of the Appell polynomials <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1673_Article_IEq1.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^{(\alpha )}_{\lambda }=\{P^{(\alpha )}_{n, \lambda }, n\in \mathbb {N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> associated with the infinite-dimensional degenerate Poisson measure <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1673_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu ^{(\alpha )}_{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>μ</mi> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. This paper presents novel properties of the kernels <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1673_Article_IEq3.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$P^{(\alpha )}_{n, \lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> in infinite dimensions, offering valuable insights, broadening the scope of our results.</p>

Falling Operators with Their Connections with the Degenerate Poisson Appell Polynomials

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发表日期：

2025年3月22日

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Anis Riahi,
Mohamed Rhaima,
Hamza Ghoudi

摘要

In this work, we derive combinatorial properties of the degenerate falling operators, present their connections with a generalization of the Appell polynomials $\mathbb {P}^{(\alpha )}_{\lambda }=\{P^{(\alpha )}_{n, \lambda }, n\in \mathbb {N}\}$ $P_{λ}^{(α)} = {P_{n, λ}^{(α)}, n \in N}$ associated with the infinite-dimensional degenerate Poisson measure $\mu ^{(\alpha )}_{\lambda }$ $μ_{λ}^{(α)}$ . This paper presents novel properties of the kernels $P^{(\alpha )}_{n, \lambda }$ $P_{n, λ}^{(α)}$ in infinite dimensions, offering valuable insights, broadening the scope of our results.