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11: 17.17 Physical Applications
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►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
►Quantum groups also apply -series extensively.
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12: 17.18 Methods of Computation
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►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.
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►Shanks (1955) applies such methods in several -series problems; see Andrews et al. (1986).
13: George E. Andrews
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►An expert on -series, he is the author of
-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.
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14: 13.24 Series
§13.24 Series
►§13.24(i) Expansions in Series of Whittaker Functions
►For expansions of arbitrary functions in series of 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 , then the quotient 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
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► 190, American Mathematical Society, 1995; Special Functions, -Series and Related Topics (with D.
… 14, American Mathematical Society, 1997;
-Series from a Contemporary Perspective (with D.
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►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.
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17: 6.6 Power Series
18: 36.15 Methods of Computation
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►
§36.15(i) Convergent Series
►Close to the origin 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 , 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
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