Bradley's Theorem

Let

$$S(\alpha,\beta,m;z)\equiv m\sum_{j=0}^\infty {\Gamma(m+j(z+1... ...amma(m+jz+1)\Gamma(\alpha+\beta+1+j(z+1))}{(\alpha)+j\over j!}$$

and $\alpha$ be a Negative Integer. Then

$$S(\alpha,\beta,m;z)={\Gamma(\beta+1-m)\over\Gamma(\alpha+\beta+1-m)},$$

where $\Gamma(z)$ is the Gamma Function.

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 346-348, 1994.

Bradley, D. ``On a Claim by Ramanujan about Certain Hypergeometric Series.'' Proc. Amer. Math. Soc. 121, 1145-1149, 1994.