<p>In this paper, we used algebraic calculus to characterize all generating functions of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2864_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$A(t)F(xtA(t)-R(t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>t</mi> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the classical orthogonal polynomials. We also applied the derivative operator to get generating functions of the form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2864_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="164" /> </InlineMediaObject> <EquationSource Format="TEX">$A(t)^kG(xtA(t)-R(t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>t</mi> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>R</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for these polynomials. Particularly, we obtained bilateral generating functions between Hermite and associate Hermite polynomials and between ultraspherical and Chebyshev polynomials of the second kind.</p>

On a Class of Rainville Type Generating Functions for Classical Orthogonal Polynomials

期刊论文
发表日期：

2025年5月30日

查看全文

Mohammed Brahim Zahaf,
Mohammed Mesk

摘要

In this paper, we used algebraic calculus to characterize all generating functions of the form $A(t)F(xtA(t)-R(t))$ $A (t) F (x t A (t) - R (t))$ for the classical orthogonal polynomials. We also applied the derivative operator to get generating functions of the form $A(t)^kG(xtA(t)-R(t))$ $A {(t)}^{k} G (x t A (t) - R (t))$ for these polynomials. Particularly, we obtained bilateral generating functions between Hermite and associate Hermite polynomials and between ultraspherical and Chebyshev polynomials of the second kind.