This article investigates the algebraic differential independence concerning the Euler $\Gamma$-function and the function $F$ in a certain class $\mathbb{F}$ which contains Dirichlet $\mathcal{L}$-functions, $\mathcal{L}$-functions in the extended Selberg class, or some periodic functions. We prove that the Euler $\Gamma$-function and the function $F$ cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions $\phi$ with $\rho(\phi)<1$.
Wei CHEN
,
Qiong WANG
. ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES[J]. Acta mathematica scientia, Series B, 2020
, 40(4)
: 1035
-1044
.
DOI: 10.1007/s10473-020-0411-3
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