On a Claim of Ramanujan About Certain Hypergeometric Series

Abstract. We state and prove a claim of Ramanujan. As a consequence, a large new class of Saalschützian hypergeometric series is summed in closed form. Hypergeometric series form an important part of the area of mathematics known as special functions. To name some applications, hypergeometric series arise in series solutions to differential equations, and in rapidly convergent expansions for $\pi$. In addition, the theory of hypergeometric series provides a unified approach to the problem of establishing identities involving binomial coefficient summations. In this theory, it is important to have closed form sums for as many series as possible. Ramanujan's enigmatic claim, which we state and prove, can be specialized to a claim about an infinite class of Saalchützian hypergeometric series. Our result contributes to the theory of hypergeometric series by providing a closed form sum for this class.