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<p>In this paper, based on the importance of Ramanujan’s Master Theorem and the <i>q</i>-beta integral, we deduce a generalized Ramanujan’s <i>q</i>-beta integral involving double Cauchy polynomials via the iteration method and investigate related expansions of Ramanujan’s Master Theorem. In addition, we gain a recurring integral formula of the generalized Ramanujan’s <i>q</i>-beta integral. More over, we consider Rogers–Szegö polynomials and their generating functions representations by Ramanujan’s <i>q</i>-beta integrals. Meanwhile, we obtain a special case of the generalized Ramanujan’s <i>q</i>-beta integral and a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/11139_2026_1404_IEq1_HTML.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="120" Type="Linedraw" Width="25" /> </InlineMediaObject> </InlineEquation> transformation from the generalized Ramanujan’s <i>q</i>-beta integral. Finally, we give the Ramanujan’s <i>q</i>-beta integral formulas from Ramanujan’s Master Theorem and propose an open problem.</p>

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Polynomial augmentation for Ramanujan’s q-beta integral and related expansions of Ramanujan’s Master Theorem

  • 期刊论文

  • 发表日期：

    2026年5月13日

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  • Jian Cao,
  • Yue Yang

摘要

In this paper, based on the importance of Ramanujan’s Master Theorem and the q-beta integral, we deduce a generalized Ramanujan’s q-beta integral involving double Cauchy polynomials via the iteration method and investigate related expansions of Ramanujan’s Master Theorem. In addition, we gain a recurring integral formula of the generalized Ramanujan’s q-beta integral. More over, we consider Rogers–Szegö polynomials and their generating functions representations by Ramanujan’s q-beta integrals. Meanwhile, we obtain a special case of the generalized Ramanujan’s q-beta integral and a transformation from the generalized Ramanujan’s q-beta integral. Finally, we give the Ramanujan’s q-beta integral formulas from Ramanujan’s Master Theorem and propose an open problem.