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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

【Author in Chinese】 朱尧辰任建华

【Author】 Zhu Yaochen (Institute of Applied Mathematics, Academia Sinica) Ren Jianhua (Department of Mathematics North-west University)

【Institution】 中国科学院应用数学研究所西北大学数学系

【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α12……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 α12,…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

【Keywords in Chinese】 有理点超越性级数引理定理超越数
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