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<p>The main aim of the present work is to give some interesting the <i>q</i>-analogues of various <i>q</i>-recurrence relations, <i>q</i>-recursion formulas, <i>q</i>-partial derivative relations, <i>q</i>-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_790_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Psi _{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ψ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_790_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Psi _{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ψ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> on two independent bases <i>q</i> and <i>p</i> of two variables, believed to be new, by using the conception of <i>q</i>-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_790_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Psi _{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ψ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_790_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Psi _{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ψ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are established when the two independent bases <i>q</i> and <i>p</i> are equal.</p>

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Certain new formulas for bibasic Humbert hypergeometric functions \(\Psi _{1}\) and \(\Psi _{2}\)

  • 期刊论文

  • 发表日期：

    2025年5月10日

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  • Ayman Shehata

摘要

The main aim of the present work is to give some interesting the q-analogues of various q-recurrence relations, q-recursion formulas, q-partial derivative relations, q-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions \(\Psi _{1}\) Ψ 1 and \(\Psi _{2}\) Ψ 2 on two independent bases q and p of two variables, believed to be new, by using the conception of q-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types \(\Psi _{1}\) Ψ 1 and \(\Psi _{2}\) Ψ 2 are established when the two independent bases q and p are equal.