In this paper, we give the algebraic independence measures for the values ofMahler type functions in complex number field and p-adic number field, respectively.
By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transc...By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given.展开更多
Let f（x）be a continued fraction with elements a<sub>n</sub>x,where coefficients a<sub>n</sub> are positive alge- braic numbers.Using the criterion of[1]for any nonzero real algebraic numbers ...Let f（x）be a continued fraction with elements a<sub>n</sub>x,where coefficients a<sub>n</sub> are positive alge- braic numbers.Using the criterion of[1]for any nonzero real algebraic numbers α<sub>1</sub>,...,α<sub>s</sub> with distinct absolute values the algebraic independence of the values f（α<sub>1</sub>）,...,f（α<sub>s</sub>）is proved under certain as- sumption concerning only with a<sub>n</sub>.For some transcendental numbers ζ the algebraic independence of values f（ζ<sup>j</sup>）（j∈Z）is also established.展开更多
The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.
In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤...In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.展开更多
In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.
1.IntroductionWe denote the set of all positive integers by N,and the fields of all rational,algebraic numbers by Q,A,respectively.Let Z~* [Z1,…,Z2] be the set of all non-zero polynomials in variables Z1,…,Z2,with i...1.IntroductionWe denote the set of all positive integers by N,and the fields of all rational,algebraic numbers by Q,A,respectively.Let Z~* [Z1,…,Z2] be the set of all non-zero polynomials in variables Z1,…,Z2,with integer coefficients.For α∈A we denotemaxand multiply from i=1 to α~((i)) by and N(a),respectively;where d is the degree of α,and展开更多
This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class F which contains Dirichlet L-functions,L-functions in the extended Selberg class...This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class F which contains Dirichlet L-functions,L-functions in the extended Selberg class, or some periodic functions. We prove that the EulerΓ-function and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions Ø with ρ(Ø) < 1.展开更多
In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma functionГ.More precisely,if P is a poly...In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma functionГ.More precisely,if P is a polynomial of n+1 variables in C[X,Y0,…,Yn-1]such that P(s,Г(s+a0),…,Г(s+an-1))≡0 for some(a0,…,an-1)∈Cn and ai-aj■Z for any 0≤i<j≤n-1,then they have P≡0.Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by H?lder in 1886,and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.展开更多
In the present note the algebraic independence of values of several Mahler series with certain particular and different parameters at algebraic points is established by means of a typical approximation method handling...In the present note the algebraic independence of values of several Mahler series with certain particular and different parameters at algebraic points is established by means of a typical approximation method handling the Liouville type series.展开更多
基金Supported by the Natural Science Foundation of Henan University（05ZDZR001）
文摘In this paper, we give the algebraic independence measures for the values ofMahler type functions in complex number field and p-adic number field, respectively.
基金Subject supported by the National Natural Science Foundation of China
文摘By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given.
基金Supported by the National Natural Science Foundation of China
文摘Let f（x）be a continued fraction with elements a<sub>n</sub>x,where coefficients a<sub>n</sub> are positive alge- braic numbers.Using the criterion of[1]for any nonzero real algebraic numbers α<sub>1</sub>,...,α<sub>s</sub> with distinct absolute values the algebraic independence of the values f（α<sub>1</sub>）,...,f（α<sub>s</sub>）is proved under certain as- sumption concerning only with a<sub>n</sub>.For some transcendental numbers ζ the algebraic independence of values f（ζ<sup>j</sup>）（j∈Z）is also established.
文摘The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.
文摘In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.
文摘In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.
文摘1.IntroductionWe denote the set of all positive integers by N,and the fields of all rational,algebraic numbers by Q,A,respectively.Let Z~* [Z1,…,Z2] be the set of all non-zero polynomials in variables Z1,…,Z2,with integer coefficients.For α∈A we denotemaxand multiply from i=1 to α~((i)) by and N(a),respectively;where d is the degree of α,and
基金by Basic and Advanced Research Project of CQCSTC(cstc2019jcyj-msxmX0107)Fundamental Research Funds of Chongqing University of Posts and Telecommunications(CQUPT:A2018-125).
文摘This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class F which contains Dirichlet L-functions,L-functions in the extended Selberg class, or some periodic functions. We prove that the EulerΓ-function and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions Ø with ρ(Ø) < 1.
基金supported by the National Natural Science Foundation of China(No.11901311)。
文摘In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma functionГ.More precisely,if P is a polynomial of n+1 variables in C[X,Y0,…,Yn-1]such that P(s,Г(s+a0),…,Г(s+an-1))≡0 for some(a0,…,an-1)∈Cn and ai-aj■Z for any 0≤i<j≤n-1,then they have P≡0.Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by H?lder in 1886,and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.
基金This work is supported by the National Natural Science Foundation of China
文摘In the present note the algebraic independence of values of several Mahler series with certain particular and different parameters at algebraic points is established by means of a typical approximation method handling the Liouville type series.
