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17 q-Hypergeometric and Related FunctionsComputation

§17.18 Methods of Computation

For computation of the q-exponential function see Gabutti and Allasia (2008).

The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities.

Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. Shanks (1955) applies such methods in several q-series problems; see Andrews et al. (1986).