Special Functions

In this chapter, we collect together those results in the unorganized portions of the second and third notebooks that pertain to special functions. The first ten entries concern the gamma function. All ten entries are either known results or can easily be derived from standard theorems on the gamma function. The next four theorems arise from the theory of Bessel functions. These four theorems also are either classical or can be simply deduced from standard results on Bessel functions. The last section of the chapter is devoted to hypergeometric functions. By far, the most interesting result is contained in Section 15. Here Ramanujan offers a tantalizingly incomplete statement about a class of Saalschützian hypergeometric series. D. Bradley [1] has provided what is probably the best theorem that can be deduced from Ramanujan’s enigmatic statement. In particular, a large, new class of Saal-schützian series is summed in closed form. We complete this chapter by listing Ramanujan’s series for l/π, arising from certain hypergeometric functions.