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11: 17.17 Physical Applications
See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. Quantum groups also apply q -series extensively. …
12: 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).
13: George E. Andrews
An expert on q -series, he is the author of q -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. …He is now collaborating with Bruce Berndt on a series of volumes explicating the brilliant and sometimes enigmatic ideas in this notebook. …
14: 13.24 Series
§13.24 Series
§13.24(i) Expansions in Series of Whittaker Functions
For expansions of arbitrary functions in series of M κ , μ ( z ) functions see Schäfke (1961b).
§13.24(ii) Expansions in Series of Bessel Functions
For other series expansions see Prudnikov et al. (1990, §6.6). …
15: 27.7 Lambert Series as Generating Functions
§27.7 Lambert Series as Generating Functions
Lambert series have the form …If | x | < 1 , then the quotient x n / ( 1 x n ) is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: …
16: Mourad E. H. Ismail
 190, American Mathematical Society, 1995; Special Functions, q -Series and Related Topics (with D. … 14, American Mathematical Society, 1997; q -Series from a Contemporary Perspective (with D. … Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …
17: 6.6 Power Series
§6.6 Power Series
The series in this section converge for all finite values of x and | z | .
18: 36.15 Methods of Computation
§36.15(i) Convergent Series
Close to the origin 𝐱 = 𝟎 of parameter space, the series in §36.8 can be used. … This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
19: 5.23 Approximations
§5.23(ii) Expansions in Chebyshev Series
Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of Γ ( 1 + x ) , 1 / Γ ( 1 + x ) , Γ ( x + 3 ) , ln Γ ( x + 3 ) , ψ ( x + 3 ) , and the first six derivatives of ψ ( x + 3 ) for 0 x 1 . …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for Γ ( 1 + x ) and its reciprocal for 0 x 1 . …
20: 34.13 Methods of Computation
For 9 j symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …