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Chapter 16 of Ramanujan’s second notebook: Theta-functions and q-series. (English) Zbl 0565.33002
Mem. Am. Math. Soc. 315, 85 p. (1985).
A basic hypergeometric series has the form $$\sum c_ n$$ with $$c_{n+1}/c_ n=rational$$ function of $$q^ n$$ for a fixed q, $$| q| <1$$. The series can be either a bilateral sum of -$$\infty$$ to $$\infty$$ or a one-sided sum from 0 to $$\infty$$. The first half of this chapter contains Ramanujan’s version of the elementary part of this subject. A number of results were new when Ramanujan found them. These include his $${}_ 1\psi_ 1$$ sum, which can be thought of as an extension of the beta integral on [0,$$\infty]$$, and contains within it Jacobi’s partial fractions for Jacobi elliptic functions, an integral version of the $${}_ 1\psi_ 1$$ sum, some continued fractions, and a limiting case of Watson’s extension of Whipple’s transformation. It is well known that this gives a proof of the Rogers-Ramanujan identities, and it is also known that Ramanujan rediscovered these identities but did not know how to prove them before seeing a paper of Rogers at least five years after writing this Notebook. Thus Ramanujan had a few blind spots, as we all do. The second half of this memoir deals with theta functions, and it must be read to be believed. Using the $${}_ 1\psi_ 1$$ sum and elementary mathematics known to freshman calculus students, Ramanujan discovered many of the classical identities of theta functions in one variable and some new results. One is $\frac{e^{-2\pi /5}}{1+}\frac{e^{-2\pi}}{1+}\frac{e^{- 4\pi}}{1+...}=\sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5+1}}{2}.$ Hardy said about this result that was contained in Ramanujan’s first letter to Hardy that it must be true, for no one could have the imagination to invent it. Berndt’s proof of the $${}_ 1\psi_ 1$$ sum is particularly interesting and illuminating.
 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals 05A19 Combinatorial identities, bijective combinatorics 11P81 Elementary theory of partitions