Summation of some infinite series by the methods of Hypergeometric functions and partial fractions

Abstract

In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: ₈F₇[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], ₅F₄[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], ₅F₄[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], ₅F₄[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], ₅F₄[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; -1], ₄F₃[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], ₄F₃[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], ₄F₃[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and ₄F₃[1, 1, 1, 1; 3, 3, 3; -1].