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Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in \( {\mathbb {C}}^2 \)

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Abstract

An equation is called a complex partial differential-difference equation, if this equation includes partial derivatives, shifts, or differences of complex-valued functions f, which can be called PDDE for short and a functional equation of the form \( f^n+g^n=1 \), where n is an integer, is called the Fermat-type equation. In this paper, we study the existence and the precise form of finite order transcendental entire solutions of several Fermat-type PDDEs and quadratic trinomial functional equations in \({\mathbb {C}}^2\). The main results of the paper generalize several existing results in this direction. In addition, we exhibited by several examples that our results are precise to some extent.

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Acknowledgements

The authors wish to thank the anonymous referees for their helpful suggestions and comments to enhance the clarity and presentation of the paper. The corresponding author is supported by “JU Research Grant” no.: S-3/10/22, dated: \( \frac{15}{17} \) \( .03.2022 \), Jadavpur University, West Bengal, India

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  1. Department of Mathematics, Malda College, Malda, 732101, West Bengal, India

    Goutam Haldar

  2. Department of Mathematics, Jadavpur University, Kolkata, 700032, West Bengal, India

    Molla Basir Ahamed

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Haldar, G., Ahamed, M.B. Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in \( {\mathbb {C}}^2 \). Anal.Math.Phys. 12, 113 (2022). https://doi.org/10.1007/s13324-022-00722-5

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