基于Tri-Gamma函数的Gamma函数的一个渐近公式
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摘要
首先,建立一个基于Tri-Gamma函数的Gamma函数的一个渐近公式.然后,得到其最佳常数、渐近展式和关于Gamma函数的双向不等式.
An asymptotic formula of the Gamma function in terms of Tri-Gamma function is established firstly. Then, the best constant, asymptotic expansion and two-sided inequalities about Gamma function are obtained.
An asymptotic formula of the Gamma function in terms of Tri-Gamma function is established firstly. Then, the best constant, asymptotic expansion and two-sided inequalities about Gamma function are obtained.
引文
[1] SRINIVASAN G K.The Gamma function:an eclectic tour[J].American Mathematics Monthly,2007,114(4):297-315.
[2] ALZER H,BATIR N.Monotonicity properties of the Gamma function[J].Appl Math Lett,2007,20(7):778-781.
[3] MORTICI C.New approximations of the Gamma function in terms of the Digamma function[J].Applied Mathematics Letters,2010,23(1):97-100.
[4] SEVLI H,BATIR N.Complete monotonicity results for some functions involving the Gamma and Polygamma functions[J].Mathematical and Computer Modelling,2011,53(9):1771-1775.
[5] QI Feng.A completely monotonic function involving the Gamma and Tri-Gamma functions[J].Kuwait Journal of Science,2016,43(6):32-40.
[6] MORTICI C,QI Feng.Asymptotic formulas and inequalities for the Gamma function in terms of the Tri-Gamma function[J].Results in Mathematics,2015,67(3/4):395-402.
[7] YOU Xu,HAN Min.Continued fraction approximation for the Gamma function based on the Tri-Gamma function[J].Journal of Mathematical Analysis and Applications,2018,457(1):389-395.
[8] SUN Bangcheng,LIU Zhiming,LI Qiang,et al.The monotonicity and convexity of a function involving Psi function with applications[J].Journal of Inequalities and Applications,2016,2016:151.
[9] ALZER H.Inequalities for the Gamma and Polygamma functions[J].Abh Math Sem Univ Hamburg,1998,68(1):363-372.
[10] CHEN Chaoping,QI Feng,SRIVASTAVA H M.Some properties of functions related to the Gamma and Psi functions[J].Integral Transforms and Special Functions,2010,21(2):153-164.
[11] CHEN Chaoping,BATIR N.Some inequalities and monotonicity properties associated with the Gamma and Psi functions[J].Applied Mathematics and Computation,2012,218(17):8217-8225.
[12] LUKE Y L.The special functions and their approximations[M].London:Academic Press,1969.
[13] CHEN Chaoping,ELEZOVIC N.Asymptotic expansions and completely monotonic functions associated with the Gamma,Psi and Polygamma functions[J].Applied Mathematics and Computation,2015,269:232-241.
[14] CHEN Chaoping,MORTICI C.New sequence converging towards the Euler-Mascheroni constant[J].Computers and Mathematics with Applications,2012,64(4):391-398.
[15] LU Dawei.Some new improved classes of convergence towards Euler’s constant[J].Applied Mathematics and Computation,2014,243:24-32.
[2] ALZER H,BATIR N.Monotonicity properties of the Gamma function[J].Appl Math Lett,2007,20(7):778-781.
[3] MORTICI C.New approximations of the Gamma function in terms of the Digamma function[J].Applied Mathematics Letters,2010,23(1):97-100.
[4] SEVLI H,BATIR N.Complete monotonicity results for some functions involving the Gamma and Polygamma functions[J].Mathematical and Computer Modelling,2011,53(9):1771-1775.
[5] QI Feng.A completely monotonic function involving the Gamma and Tri-Gamma functions[J].Kuwait Journal of Science,2016,43(6):32-40.
[6] MORTICI C,QI Feng.Asymptotic formulas and inequalities for the Gamma function in terms of the Tri-Gamma function[J].Results in Mathematics,2015,67(3/4):395-402.
[7] YOU Xu,HAN Min.Continued fraction approximation for the Gamma function based on the Tri-Gamma function[J].Journal of Mathematical Analysis and Applications,2018,457(1):389-395.
[8] SUN Bangcheng,LIU Zhiming,LI Qiang,et al.The monotonicity and convexity of a function involving Psi function with applications[J].Journal of Inequalities and Applications,2016,2016:151.
[9] ALZER H.Inequalities for the Gamma and Polygamma functions[J].Abh Math Sem Univ Hamburg,1998,68(1):363-372.
[10] CHEN Chaoping,QI Feng,SRIVASTAVA H M.Some properties of functions related to the Gamma and Psi functions[J].Integral Transforms and Special Functions,2010,21(2):153-164.
[11] CHEN Chaoping,BATIR N.Some inequalities and monotonicity properties associated with the Gamma and Psi functions[J].Applied Mathematics and Computation,2012,218(17):8217-8225.
[12] LUKE Y L.The special functions and their approximations[M].London:Academic Press,1969.
[13] CHEN Chaoping,ELEZOVIC N.Asymptotic expansions and completely monotonic functions associated with the Gamma,Psi and Polygamma functions[J].Applied Mathematics and Computation,2015,269:232-241.
[14] CHEN Chaoping,MORTICI C.New sequence converging towards the Euler-Mascheroni constant[J].Computers and Mathematics with Applications,2012,64(4):391-398.
[15] LU Dawei.Some new improved classes of convergence towards Euler’s constant[J].Applied Mathematics and Computation,2014,243:24-32.