基于欧拉函数的其他特殊函数完全单调性证明
发布时间：2018-09-12 12:13
【摘要】：完全单调性以及对数完全单调性是Gamma函数的重要性质。提出对基于欧拉函数的其他特殊函数完全单调性进行证明与研究。主要证明了一些包含Gamma函数的特殊函数完全单调性,推导一些重要的不等式。首先,利用整数环理论的单位群和理论本身性质,对特殊函数的互素性质进行描述,通过给出整数环素元的形式,以及整数素元表达式和部分非整数素元表达式,并根据整数环商环的性质利用同构映射方式分析特殊函数的互质性质;其次,在包含Gamma函数的特殊函数互质性质下,根据欧拉函数的性质对包含Gamma函数的特殊函数完全单调性质进行证明,任意欧拉函数在一定区间内的各阶导数为正整数的条件,在任意欧拉函数的正函数也满足正整数的情况下,即证明特殊函数在一定区间内为完全单调函数。
[Abstract]:Complete monotonicity and logarithmic complete monotonicity are important properties of Gamma functions. The complete monotonicity of other special functions based on Euler function is proved and studied. The complete monotonicity of some special functions including Gamma functions is proved and some important inequalities are derived. First of all, by using the unit group of integer ring theory and the properties of the theory itself, the coprime properties of special functions are described. The form of integer ring prime element, the expression of integer prime element and the expression of partial non-integer prime element are given. According to the properties of quotient rings of integer rings, the mutual prime properties of special functions are analyzed by means of isomorphic mapping. Secondly, under the properties of special functions containing Gamma functions, According to the properties of Euler function, the complete monotone property of special function containing Gamma function is proved. The condition that every order derivative of any Euler function is a positive integer in a certain interval is proved. If the positive function of any Euler function also satisfies the positive integer, it is proved that the special function is completely monotone in a certain interval.
【作者单位】： 黄河交通学院基础教学部;
【基金】：河南省政府决策研究招标课题(2015B151)
【分类号】：O174.6
本文编号：2238970
[Abstract]:Complete monotonicity and logarithmic complete monotonicity are important properties of Gamma functions. The complete monotonicity of other special functions based on Euler function is proved and studied. The complete monotonicity of some special functions including Gamma functions is proved and some important inequalities are derived. First of all, by using the unit group of integer ring theory and the properties of the theory itself, the coprime properties of special functions are described. The form of integer ring prime element, the expression of integer prime element and the expression of partial non-integer prime element are given. According to the properties of quotient rings of integer rings, the mutual prime properties of special functions are analyzed by means of isomorphic mapping. Secondly, under the properties of special functions containing Gamma functions, According to the properties of Euler function, the complete monotone property of special function containing Gamma function is proved. The condition that every order derivative of any Euler function is a positive integer in a certain interval is proved. If the positive function of any Euler function also satisfies the positive integer, it is proved that the special function is completely monotone in a certain interval.
【作者单位】： 黄河交通学院基础教学部;
【基金】：河南省政府决策研究招标课题(2015B151)
【分类号】：O174.6
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