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<p>In our paper with M. E. H. Ismail [<CitationRef CitationID="CR7">7</CitationRef>], we are interested in studying the zeros of a family of entire functions introduced by Ramanujan, using contour integrals and connection formulas for solutions of linear <i>q</i>-difference equations. Many years ago, M. E. H. Ismail provided a simple proof of Ramanujan’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(_1\psi _1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> <msub> <mi>ψ</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> summation formula [<CitationRef CitationID="CR6">6</CitationRef>]. In this note, we aim to present an analytic proof of this formula using similar approaches to those in [<CitationRef CitationID="CR7">7</CitationRef>]. Furthermore, building on these methods, we derive a decomposition formula for the sum-function of a <i>q</i>-analog of the Euler series, which has the potential to be generalized to a broader class of <i>q</i>-difference equations.</p>

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A note on contour integrals representing Jackson integrals and their applications

  • 期刊论文

  • 发表日期：

    2025年11月7日

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  • Changgui Zhang

摘要

In our paper with M. E. H. Ismail [7], we are interested in studying the zeros of a family of entire functions introduced by Ramanujan, using contour integrals and connection formulas for solutions of linear q-difference equations. Many years ago, M. E. H. Ismail provided a simple proof of Ramanujan’s \(_1\psi _1\) 1 ψ 1 summation formula [6]. In this note, we aim to present an analytic proof of this formula using similar approaches to those in [7]. Furthermore, building on these methods, we derive a decomposition formula for the sum-function of a q-analog of the Euler series, which has the potential to be generalized to a broader class of q-difference equations.