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# On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

## Sur les solutions des equations de type Fermat quadratiques dans ℂ2 engendrées par des opérateurs différentiels C-SHIFT d’ordre un

Abhijit Banerjee
University of Kalyani
India

Jhuma Sarkar
University of Kalyani
India

Published on 5 January 2024   DOI : 10.21494/ISTE.OP.2023.1049

### Mots-clés

This article is devoted to explore various forms of transcendental entire solution of different quadratic trinomials generated by first order linear c-shift operator. We also investigate the forms of solutions of certain quadratic trinomials under linear and mixed partial differential operators. Our paper improves the results of Li-Xu [Axioms, 126(10)(2021), 1-19] in two directions. In addition, in a corollary, deducted from one of our main result, we extend a result of Zhang et al. [ Aims Math., 7(2022), 11597-11613]. A series of examples have been exhibited to justify the existence and forms of transcendental entire solution of such equations. In the last section of the paper we have put a relevant question for future research.

This article is devoted to explore various forms of transcendental entire solution of different quadratic trinomials generated by first order linear c-shift operator. We also investigate the forms of solutions of certain quadratic trinomials under linear and mixed partial differential operators. Our paper improves the results of Li-Xu [Axioms, 126(10)(2021), 1-19] in two directions. In addition, in a corollary, deducted from one of our main result, we extend a result of Zhang et al. [ Aims Math., 7(2022), 11597-11613]. A series of examples have been exhibited to justify the existence and forms of transcendental entire solution of such equations. In the last section of the paper we have put a relevant question for future research.