(1) These coefficients are expressed by the hypergeometric functions and a method with high efficiency in computations in presented.

(2) Efficient calculating method for zonal polynomials is the key of good approximation for hypergeometric function with matrix argument.

(3) By using a simple algorithm for the summation of basic hypergeometric series, summation formulas for some basic hypergeometric series are obtained.

(4) The Six Sigma Black Belt should be familiar with the commonly used probability distributions, including: hypergeometric, binomial, Poisson, normal, exponential, chi-square, Student's t, and F.

(5) The Six Sigma Black Belt should be familiar with the commonly used probability, including: hypergeometric, binomial, Poisson, normal, exponential, chi-square, Student's t, and F.

(6) A class of discrete type random variable probability distributions, called negative geometric distribution and negative hypergeometric distribution, are discussed.

(7) In this paper, the inclination functions are expressed as the hypergeometric functions.

(8) With the boundary conditions of bound states, we have obtained the corresponding energy spectrum via an expression and wave functions in terms of hypergeometric functions.

(9) In Chapter 2, we introduce the basic knowledge of the basic hypergeometric series, including basic concepts, some summation formulas, transformation formulas, etc.

(10) Furthermore, some new Rogers-ramanujan type identities are obtained by a transformation of basic hypergeometric series.

(11) The main contents of this thesis can be summarized as follows:In Chapter 1, we introduce the development of the basic hypergeometric series.

(12) This dissertation studies the applications of the inversion techniques and its equivalent form in finding and proving the hypergeometric series identities.

(13) The zeroth order approximation of the solution can be expressed in terms of confluent hypergeometric functions.

(14) The radial bound state solutions are expressed in terms of the confluent hypergeometric functions and the energy equation is derived from the boundary condition satisfied by the radial wavefunctions.

(15) In Chapter 1, we first look back the history of hypergeometric series and basic hypergeometric series, and then introduce some necessary concepts and notations.

(16) In part II, we consider the recurrence formula of double hypergeometric terms.