Some Applications Of Dougall's ₅F₄ Summation

With the help of some properties of the gamma function and Dougall's ₅F₄ summation formula for the classical hypergeometric series,by differentiating the Dougall's ₅F₄ summation formula with respect to parameter a,to get some new series identities.Then,we use the corresponding parameter transformation to obtain some series expansion formulae of four parameter,which can produce infinitely many Ramanujan type series for 1/? and Ramanujan type series for some other constants.The whole dissertation contains four chapter.Chapter 1,we define the gamma function,digamma function,trigamma function and discuss some of their properties.Besides we recommend the funda-mental knowledge about hypergeometric functions with some necessary identities and their recent research.Finally,we express the main results of this dissertation.Chapter 2,by differentiating the Dougall's ₅F₄ summation formula with re-spect to parameter a,we obtain the main results.Then,we have applications toward some example about Ramanujan type series for 1/?.In Chapter 3,we presenl some general series expansion for ?2,?-3?2/3?and?1/?????-2?3/4?,contains examples for respectively each type.In Chapter 4,we will express Ramanujan type series for 1/?2 contains some theories and its applications.