The class of meromorphic functions sharing values with their difference polynomials

Molla Basir Ahamed ()
Additional contact information
Molla Basir Ahamed: Jadavpur University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 4, 1158-1169

Abstract: Abstract The field of c-periodic meromorphic functions in $$ \mathbb {C} $$ C is defined by $$ {\mathcal {M}}_c:=\{f : f\; \text{ is } \text{ meromorphic } \text{ in }\; \mathbb {C}\;\text{ and }\; f(z+c)=f(z)\} $$ M c : = { f : f is meromorphic in C and f ( z + c ) = f ( z ) } and the c-shift linear difference polynomial of a meromorphic function f is defined by $$\begin{aligned} L^n_c(f)=a_nf(z+nc)+\cdots +a_1f(z+c)+a_0f(z), \end{aligned}$$ L c n ( f ) = a n f ( z + n c ) + ⋯ + a 1 f ( z + c ) + a 0 f ( z ) , where $$ a_n(\ne 0), \ldots , a_1, a_0\in \mathbb {C} $$ a n ( ≠ 0 ) , … , a 1 , a 0 ∈ C . It is easy to see that if $$ a_j=\left( {\begin{array}{c}n\\ j\end{array}}\right) (-1)^{n-j} $$ a j = n j ( - 1 ) n - j , then $$ L^n_c(f)=\Delta ^n_cf $$ L c n ( f ) = Δ c n f , where $$ \Delta ^n_cf $$ Δ c n f is a higher difference operator of f. Let $$\begin{aligned} {\mathcal {S}}_c=\{f: f \; \text{ is } \text{ meromorphic } \text{ in } \; \mathbb {C}\;\text{ and }\; L^n_c(f)\equiv f\}. \end{aligned}$$ S c = { f : f is meromorphic in C and L c n ( f ) ≡ f } . In this paper, we study the value sharing problem between a meromorphic functions f and their linear difference polynomials $$ L^n_c(f) $$ L c n ( f ) and prove a result generalizing several existing results. In addition, we find the class $$ {\mathcal {S}}_c $$ S c completely which gives the positive answers to a conjecture and an open problem in this direction.

Keywords: Uniqueness problem; Linear difference polynomials; Meromorphic functions; General solutions; Shared values; 30D35 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-022-00329-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:54:y:2023:i:4:d:10.1007_s13226-022-00329-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-022-00329-3

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().