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<p>In this work, we introduce certain infinite order integro-differential operators associated with Dunkl operator and study the Watson-type integral transforms which provide the necessary and sufficient conditions for a class of infinite order integro-differential operators being unitary on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\mu _\alpha )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also give a Plancherel-type theorem for the operators mentioned above. Furthermore, solutions of some classes of related integro-differential equations are also considered.</p>

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The Infinite Order Integro-Differential Operators Associated with Dunkl Operator on \(\mathbb {R}\)

  • 期刊论文

  • 发表日期：

    2026年3月11日

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  • Randhir Kumar Verma,
  • Akhilesh Prasad

摘要

In this work, we introduce certain infinite order integro-differential operators associated with Dunkl operator and study the Watson-type integral transforms which provide the necessary and sufficient conditions for a class of infinite order integro-differential operators being unitary on \(L^2(\mu _\alpha )\) L 2 ( μ α ) . We also give a Plancherel-type theorem for the operators mentioned above. Furthermore, solutions of some classes of related integro-differential equations are also considered.