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

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

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

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).

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

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/Abstract

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.

参考文献

[1] Bank S, Kaufman R. An extension of Hölders theorem concerning the Gamma function. Funkcial Ekvac, 1976, 19:53-63
[2] Cimpoea M, Nicolae F. Independence of Artin L-functions. Forum Mathematicum, 2019, 31(2):529-534
[3] Chiang Y M, Feng S J. Difference independence of the Riemann zeta function. Acta Arithmetica, 2006, 125(4):317-329
[4] Gross F, Osgood C F. A generalization of the nielson-holder theorem about Γ(z). Complex Variables Theory and Application:An International Journal, 1998, 37:243-250
[5] Han Q. Liu J B, Wang Q Y, et al. algebraic differential independence regarding the Euler Γ function and the riemann ζ function. arXIV:1811.04188v1
[6] Han Q, Liu J B. On differrntial independence of ζ and Γ. arXIV:1811.09336v2
[7] Hausdorff F. Zum Hölderschen Satz über ζ(z). Math Ann, 1925, 94:244-247
[8] Hilbert D. Mathematische probleme. Arch Math Phys, 1901, 63(44):213-317
[9] Hilbert D. Mathematical problems. Bull Amer Math Soc (NS), 2000, 37(4):407-436
[10] Hölder O. Über die Eigenschaft der Γ-Function, keiner algebraischen Differentialgleichung zu genügen. Math Ann, 1887, 28:1-13
[11] Karatsuba A A, Voronin S M. The Riemann Zeta Function. Berlin:Walter de Gruyter, 1992
[12] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin:Walter de Gruyter, 1993
[13] Li B Q, Ye Z. Algebraic differential equations with functional coefficients concerning ζ and Γ. J Differential Equations, 2016, 260:1456-1464
[14] Li B Q, Ye Z. On differential independence of the Riemann zetafunction and the Eulergamma function. Acta Arith, 2008, 135(4):333-337
[15] Li B Q, Ye Z. Algebraic differential equations concerning the Riemann zeta function and the Euler gamma function. Indiana Univ Math J, 2010, 59:1405-1415
[16] Li B Q, Ye Z. On algebraic differential properties of the Riemann ζ-function and the Euler Γ-function. Complex Var Elliptic Equ, 2011, 56:137-145
[17] Liao L W, Yang C C. On some new properties of the gamma function and the Riemann zeta function. Math Nachr, 2003, 257:59-66
[18] Lü F. A study on algebraic differential equationd of gamma function and dirichlet series. J Math Anal Appl, 2018, 462(2):1195-1204
[19] Lü F. On algebraic differential equations for the gamma function and L-functions in the extendeded selberg class. Bulletin of the Australian Mathematical Society, 2017, 96:36-43
[20] Markus L. Differential independence of Γ and ζ. J Dynam Differential Equations, 2007, 19:133-154
[21] Mijajlovic Z, Malesevic B. Differentially transcendental functions. Bull Belg Math Soc Simon Stevin, 2008, 15:193-201
[22] Boltovskoi D M. On hypertranscendence of the function ξ(x, s). Izv Politekh Inst Warsaw, 1914, 2:1-16
[23] Moore E. Concerning transcendentally transcendental functions. Math Ann, 1897, 48:49-74
[24] Nevanlinna R. Analytic Functions. Berlin:Springer, 1970
[25] Ostrowski A. Über Dirichletsche Reihen und algebraische Differentialgl eichungen. Math Z, 1920, 8:241-298
[26] Ostrowski A. Neuer Beweis des Hölderschen Satzes, daβ die Gammafunktion keiner algebraischen Differentialgleichung genügt. Mathematische Annalen, 1918, 79:286-288
[27] Ostrowski A. Zum Hölderschen Satz über Γ(z). Math Annalen, 1925, 94:248-251
[28] Stadigh V E E. Ein Satz über Funktionen, die algebraische Differentialgleichungen befriedigen, und ber die Eigenschaft der Funktion ζ(s) keiner solchen Gleichung zu genügen. Diss Frenckell Dr, 1902
[29] Steuding J. Value Distribution of L-Functions. Berlin:Springer-Verlag, 2007
[30] Titchmarsh E. The Theory of Functions. 2nd ed. Oxford Univ Press, 1968
[31] Voronin S M. On the functional independence of Dirichlet L-functions. Acta Arith, 1975, 27:493-503
[32] Voronin S M. On differential independence of ζ functions. Soviet Math Dokl, 1973, 14:607-609

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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