<p>In this paper, we determine all monic orthogonal polynomial sequences <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_647_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{P_n\}_{n\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, such that the monic polynomial <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_647_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="244" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_n(x):=\xi ^{-1}_n\int _0^{+\infty }t^ne^{-t^2}P_n(xt) \ \textrm{d}t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msubsup> <mi>ξ</mi> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mo>+</mo> <mi>∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mi>n</mi> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </msup> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="4pt" /> <mtext>d</mtext> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_647_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_647_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\xi _n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ξ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is a normalisation factor) is also orthogonal. As a consequence, we deal with some integral representations involving Hermite polynomials.</p>

Legendre polynomials in terms of integrals involving Hermite polynomials

期刊论文
发表日期：

2025年5月16日

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Baghdadi Aloui

摘要

In this paper, we determine all monic orthogonal polynomial sequences $\{P_n\}_{n\ge 0}$ ${P_{n}}_{n \geq 0}$ , such that the monic polynomial $Q_n(x):=\xi ^{-1}_n\int _0^{+\infty }t^ne^{-t^2}P_n(xt) \ \textrm{d}t$ $Q_{n} (x) : = ξ_{n}^{- 1} \int_{0}^{+ \infty} t^{n} e^{- t^{2}} P_{n} (x t) d t$ , $n\ge 0$ $n \geq 0$ , ( $\xi _n$ $ξ_{n}$ is a normalisation factor) is also orthogonal. As a consequence, we deal with some integral representations involving Hermite polynomials.