ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

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ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

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doi:10.1007/s10473-020-0411-3

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (4): 1035-1044.doi: 10.1007/s10473-020-0411-3

陈玮, 王琼

School of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

收稿日期:2019-04-21 修回日期:2019-09-12 出版日期:2020-08-25 发布日期:2020-08-21
通讯作者: Qiong WANG E-mail:qiongwangsdu@126.com
作者简介:Wei CHENE-mail:weichensdu@126.com
基金资助:
This work of both authors was partially supported by Basic and Advanced Research Project of CQ CSTC (cstc2019jcyj-msxmX0107), and Fundamental Research Funds of Chongqing University of Posts and Telecommunications (CQUPT: A2018-125).

Wei CHEN, Qiong WANG

School of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received:2019-04-21 Revised:2019-09-12 Online:2020-08-25 Published:2020-08-21
Contact: Qiong WANG E-mail:qiongwangsdu@126.com
Supported by:
This work of both authors was partially supported by Basic and Advanced Research Project of CQ CSTC (cstc2019jcyj-msxmX0107), and Fundamental Research Funds of Chongqing University of Posts and Telecommunications (CQUPT: A2018-125).

摘要： 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 $\rho(\phi)<1$ .

关键词: Gamma function, $\mathcal{L}$ -functions, algebraic differential independence, algebraic differential equations

Abstract: 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 $\rho(\phi)<1$ .

Key words: Gamma function, $\mathcal{L}$ -functions, algebraic differential independence, algebraic differential equations

中图分类号:

11M36

陈玮, 王琼. ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES[J]. 数学物理学报(英文版), 2020, 40(4): 1035-1044.

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.

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