In this paper, due to the importance of q-beta integral, we generalize q-beta integral involving polynomials through iterative methods. In addition, we achieve a recurrent q-beta integral formula by transformational technique. Moreover, we get <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1890_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$U(n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> type Kalnins–Miller transformation and some <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1890_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$ _3\phi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>3</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> transformations by q-beta integral. Meanwhile, we attain a generalized q-beta integral by q-difference equation. Finally, we give an additional application of q-beta integral from the finite q-binomial theorem to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1890_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$ _3\phi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>3</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> transformation.

A Note on Generalizations of q-Beta Integral and Related Transformational Identities

期刊论文
发表日期：

2025年5月24日

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Jian Cao,
Yue Yang

摘要

In this paper, due to the importance of q-beta integral, we generalize q-beta integral involving polynomials through iterative methods. In addition, we achieve a recurrent q-beta integral formula by transformational technique. Moreover, we get $U(n+1)$ $U (n + 1)$ type Kalnins–Miller transformation and some $ _3\phi _2$ $_{3} ϕ_{2}$ transformations by q-beta integral. Meanwhile, we attain a generalized q-beta integral by q-difference equation. Finally, we give an additional application of q-beta integral from the finite q-binomial theorem to $ _3\phi _2$ $_{3} ϕ_{2}$ transformation.