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The class of meromorphic functions sharing values with their difference polynomials

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Abstract

The field of c-periodic meromorphic functions in \( \mathbb {C} \) is defined by \( {\mathcal {M}}_c:=\{f : f\; \text{ is } \text{ meromorphic } \text{ in }\; \mathbb {C}\;\text{ and }\; f(z+c)=f(z)\} \) and the c-shift linear difference polynomial of a meromorphic function f is defined by

$$\begin{aligned} L^n_c(f)=a_nf(z+nc)+\cdots +a_1f(z+c)+a_0f(z), \end{aligned}$$

where \( a_n(\ne 0), \ldots , a_1, a_0\in \mathbb {C} \). It is easy to see that if \( a_j=\left( {\begin{array}{c}n\\ j\end{array}}\right) (-1)^{n-j} \), then \( L^n_c(f)=\Delta ^n_cf \), where \( \Delta ^n_cf \) is a higher difference operator of f. Let

$$\begin{aligned} {\mathcal {S}}_c=\{f: f \; \text{ is } \text{ meromorphic } \text{ in } \; \mathbb {C}\;\text{ and }\; L^n_c(f)\equiv f\}. \end{aligned}$$

In this paper, we study the value sharing problem between a meromorphic functions f and their linear difference polynomials \( L^n_c(f) \) and prove a result generalizing several existing results. In addition, we find the class \( {\mathcal {S}}_c \) completely which gives the positive answers to a conjecture and an open problem in this direction.

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Acknowledgements

The author wish to thank the anonymous reviewer/referees for the helpful comments and suggestions to improve the clarity and presentation of the manuscript. I also wish to thank Prof. Kai Liu for the helpful suggestions on finding the precise form of the solutions to the difference equation appears in our investigation. The research work of the 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, Jadavpur University, Kolkata, West Bengal, 700032, India

    Molla Basir Ahamed

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Communicated by Kaushal Verma.

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Ahamed, M.B. The class of meromorphic functions sharing values with their difference polynomials. Indian J Pure Appl Math 54, 1158–1169 (2023). https://doi.org/10.1007/s13226-022-00329-3

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