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<p>In this paper, we introduce a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-analog of Tricomi expansion based on Hahn’s difference operator. Some properties of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-Tricomi expansion are derived and proved in terms of incomplete <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-gamma functions. Also, a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-analog of the exponential integral is presented as a series expansion of incomplete <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-gamma functions and shown to be a limiting case of a <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2005_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>,</mo> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$q,\omega $</EquationSource> </InlineEquation>-Tricomi expansion.</p>

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\(q,\omega \)-Tricomi expansions and exponential integral associated with Hahn difference operator

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  • 发表日期：

    2025年2月14日

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  • Karima M. Oraby

摘要

In this paper, we introduce a q , ω $q,\omega $ -analog of Tricomi expansion based on Hahn’s difference operator. Some properties of q , ω $q,\omega $ -Tricomi expansion are derived and proved in terms of incomplete q , ω $q,\omega $ -gamma functions. Also, a q , ω $q,\omega $ -analog of the exponential integral is presented as a series expansion of incomplete q , ω $q,\omega $ -gamma functions and shown to be a limiting case of a q , ω $q,\omega $ -Tricomi expansion.