SOME SUMMATION FORMULAS FOR THE HYPERGEOMETRIC SERIES _(r+2)F_(r+1)(1/2)

摘要

The aim of this paper is to obtain explicit expressions of the generalized hypergeometric function (_(r+2)F_(r+1)[a,b, 1/2(a+b+j+1), (f_r+m_r)(f_r);1/2] for j = 0, ±1,..., ±5, where r pairs of numeratorial and denominatorial parameters differ by positive integers m_r. The results are derived with the help of an expansion in terms of a finite sum of _2F_1(2) functions and a generalization of Gauss' second summation theorem due to Lavoie et al. [J. Comput. Appl. Math. 72, 293-300 (1996)]. Some special and limiting cases are also given.