The class of meromorphic functions sharing values with their difference polynomials

摘要

Abstract The field of c-periodic meromorphic functions in Cdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ mathbb {C} $$end{document} is defined by Mc:={f:fismeromorphicinCandf(z+c)=f(z)}documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {mathcal {M}}_c:={f : f; text{ is } text{ meromorphic } text{ in }; mathbb {C};text{ and }; f(z+c)=f(z)} $$end{document} and the c-shift linear difference polynomial of a meromorphic function f is defined by Lcn(f)=anf(z+nc)+⋯+a1f(z+c)+a0f(z),documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} L^n_c(f)=a_nf(z+nc)+cdots +a_1f(z+c)+a_0f(z), end{aligned}$$end{document}where an(≠0),…,a1,a0∈Cdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ a_n(ne 0), ldots , a_1, a_0in mathbb {C} $$end{document}. It is easy to see that if aj=nj(-1)n-jdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ a_j=left( {begin{array}{c}n jend{array}}right) (-1)^{n-j} $$end{document}, then Lcn(f)=Δcnfdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ L^n_c(f)=Delta ^n_cf $$end{document}, where Δcnfdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ Delta ^n_cf $$end{document} is a higher difference operator of f. Let Sc={f:fismeromorphicinCandLcn(f)≡f}.documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} {mathcal {S}}_c={f: f ; text{ is } text{ meromorphic } text{ in } ; mathbb {C};text{ and }; L^n_c(f)equiv f}. end{aligned}$$end{document}In this paper, we study the value sharing problem between a meromorphic functions f and their linear difference polynomials Lcn(f)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ L^n_c(f) $$end{document} and prove a result generalizing several existing results. In addition, we find the class Scdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {mathcal {S}}_c $$end{document} completely which gives the positive answers to a conjecture and an open problem in this direction.