On the Algebraic Difference Independence of the Euler Gamma Function Γ and Dirichlet Series

We study the question of the algebraic difference independence of the Euler gamma function Γ and the functions in a certain class F , which contains those Dirichlet series as L-functions in the extended Selberg class S ♯ and some periodic functions. The main results in this paper are the difference...

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Bibliographic Details
Published in:Constructive approximation 2023-08, Vol.58 (1), p.121-149
Main Authors: Li, Xiao-Min, Tahir, Hassan, Gao, Xue-Yuan
Format: Article
Language:English
Subjects:
Algebra
Analysis
Differential equations
Dirichlet problem
Gamma function
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
Periodic functions
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Summary:We study the question of the algebraic difference independence of the Euler gamma function Γ and the functions in a certain class F , which contains those Dirichlet series as L-functions in the extended Selberg class S ♯ and some periodic functions. The main results in this paper are the difference analogues of the corresponding results from Lü (J Math Anal Appl 462(2):1195–1204, 2018) that showed that the Euler gamma function Γ and the functions in F can not satisfy a class of algebraic differential equations with meromorphic coefficients ϕ of Nevanlinna’s characteristics satisfying T ( r , ϕ ) = o ( r ) , as r → ∞ . Examples are provided to show that the main results in this paper, in a sense, are best possible.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-022-09600-6