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Watson’s basic analogue of Ramanujan’s entry 40 and its generalization. (English) Zbl 0804.33015

In [“Ramanujan’s continued fraction”, Proc. Camb. Philos. Soc. 31, No. 4, 7-17 (1935; Zbl 0011.11504)], G. N. Watson gives a \(q\)-analogue of entry 40 in Ramanujan’s notebook. It is a terminating continued fraction. D. R. Masson describes in [C. R. Math. Acad. Sci., Soc. R. Can. 13, 167-172 (1991; Zbl 0766.33005)] the so called “missing companion” of Ramanujan’s entry 40. In this paper, a nonterminating version of Watson’s continued fraction as well as a \(q\)-analogue of Masson’s theorem is described. This is achieved by considering very well poised \(_ 8 \varphi_ 7\) series, which are seen as limits of terminating very well poised, balanced \(_{10}\varphi_ 9\) series. This paper derives the important continuous relation for the latter and obtains the corresponding continued fraction. A basic tool in the derivation is Pincherle’s theorem for difference equations. This has been used before in this context and it brings out the connection between several of Ramanujan’s chapter 12 entries.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
39A10 Additive difference equations
40A15 Convergence and divergence of continued fractions