bilateral series

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2: 17.18 Methods of Computation
§17.18 Methods of Computation
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. … Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).
3: 17.4 Basic Hypergeometric Functions
§17.4(iv) Classification
The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … The series (17.4.1) is said to be k-balanced when $r=s$ and …
4: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
§17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
Note that for the equations below, the constraints are included to guarantee that the infinite series representation (17.4.3) of the ${{}_{r}\psi_{r}}$ functions converges. …
Bailey’s Bilateral Summations
For similar formulas see Verma and Jain (1983).