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<p>Classical Fermat-type partial differential functional equations have been extensively studied. However, their <i>q</i>-shift analogues-particularly in the context of transcendental entire functions, remain largely unexplored. This paper aims to address this significant gap in the literature by investigating the solutions of <i>q</i>-shift partial differential functional equations, with a specific focus on Fermat-type and trinomial <i>q</i>-shift equations. We provide both theoretical insights and illustrative examples to support our analysis. Our study reveals structural properties and behaviors of the solutions to these <i>q</i>-shift variants, thereby extending the existing framework within complex analysis. The findings presented in this work may have broader implications in various branches of mathematics and mathematical physics, especially in areas where <i>q</i>-analogues play a fundamental role. These results open up new avenues for further research and potential applications.</p>

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Solutions of Two Variants q-Shift and q-Difference Partial Differential Functional Equations in \(\mathbb {C}^{2}\)

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Goutam Haldar,
  • Abhijit Banerjee,
  • Ashalata Roy,
  • Jhuma Sarkar

摘要

Classical Fermat-type partial differential functional equations have been extensively studied. However, their q-shift analogues-particularly in the context of transcendental entire functions, remain largely unexplored. This paper aims to address this significant gap in the literature by investigating the solutions of q-shift partial differential functional equations, with a specific focus on Fermat-type and trinomial q-shift equations. We provide both theoretical insights and illustrative examples to support our analysis. Our study reveals structural properties and behaviors of the solutions to these q-shift variants, thereby extending the existing framework within complex analysis. The findings presented in this work may have broader implications in various branches of mathematics and mathematical physics, especially in areas where q-analogues play a fundamental role. These results open up new avenues for further research and potential applications.