Approximations Of Ratio Of Gamma Functions And Its Applications

The ratio of gamma functions is one of the key research problems,and has been widely used in many fields.For example,Minc-Sathre quotient and Wallis ratio are calculated by the ratio of gamma functions.The ratio of gamma functions also can be used in the study of integrals of Mellin-Barnes type and of hypergeometric functions.The density functions of some distributions and quantile of distribution can be calculated using the ratio of gamma functions.In addition,the characteristic function of the logarithmic likelihood ratio test statistic can be written as the product of the ratios of gamma functions.Therefore,studying the approximation of the ratio of gamma functions is of great theoretical and practical significance.In the beginning,the ratio of gamma functions was based on the deterministic function?(x+1)/?(x+1/2)as the main research objective.With the maturity of the theory of gamma function and the improvement of research results,extending the study of ratio of gamma functions to the ratio?(x+t)/?(x+s)is to meet the practical needs of different fields.In this thesis,we study two kinds of approximations of the ratio of gamma functions:polynomial approximation and Digamma functional approximation.Each kind is divided into two types:power series type and contin-ued fraction type.With regard to the different types of approximations,this paper studies the applications in median of gamma distribution and central binomial coefficients to obtain better approximation results.In the last chapter,the approximation results of ratio of gamma functions are compared with those of predecessors by numerical calculation,and have faster convergence speed and better superiority.