Node Document
一类缺项级数在有理点上值的超越性
The Transcendence of the Values of Certain Gap Series in Rational Points
【Abstract in Chinese】 我们用N,Q分别表示全体自然数和全体有理数的集合。令σ(z)=sum from n=1 to ∞αnzCn,(1)其中αn∈Q,Cn∈N,Cn↑∞。用Mk表示α1,α2……αk的公分母。对于δ>0及a∈N,a≥2定义集合S(a,δ)={p/q|p/q∈Q,(p,q)=1,q≥a,|p|≤qδ} (2) 本文得到了两个关于σ(z)在有理点上值的超越性的判定定理: 定理1 如果对于级数(1),存在常数A>0使那么,当时,对于任何p/q∈S(a,δ),σ(p/q)是超越数。定理2 如果对于级数(1),存在无穷实数列βn(n=1,2…)适合其中k0∈N,K>0是常数。那么,当(4)、(5)、(6)成立时,对于任何p/q∈S(a,δ),σ(p/q)是超越数。
【Abstract】 We use N and Q to denote the set of all integers and of all rational numbers respectively.Putσ(z)=sum from n=1 to ∞(αnzCn),where αn∈Q, Cn∈N, Cn↑∞.Let Mk denote the common denominator of α1,α2,…αk. Furthermore, We define a setS(a,δ)={p/q | p/q∈Q,(p,q)=1,q≥a, |q|≤qδ} (2)for δ>0,a∈N and a≥2.In this paper, we have obtained two discriminating theorems about the transcendence of the values of σ(z) in rational points.They areTheorem 1 If there exists a constant A>0 for series (1) such that|αn|≤A(n=1,2,…)。 (3)Then, when(?) Ck/Ck+1=M*< 1, (4)(?) ln Mk/Ck=0 (5)δ<1-M*, (6)σ(p/q) is transcendental for any rational number p/q∈S(a,δ). Theorem 2 If there exists a infinite real number sequence βn (n=1,2,…) for series (1) satisfying|βm+n|≤K |βm| |βn| (m,n=1,2,…),|αk|≤|βk| (k≥k0),where k0∈N and K>0 is a Constant. Then,when Conditions (4),(5),(6) hold, σ(p/q) is transcendental for any rational number p/q∈S(a,δ). Back
- 【Source】 西北大学学报(自然科学版) ,Journal of Northwest University(Natural Science Edition) , Editorial E-mail ,1982(04)
- 【Downloads】13