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1: 28.15 Expansions for Small q
§28.15(i) Eigenvalues λ ν ( q )
§28.15(ii) Solutions me ν ( z , q )
2: 28.6 Expansions for Small q
§28.6(i) Eigenvalues
The coefficients of the power series of a 2 n ( q ) , b 2 n ( q ) and also a 2 n + 1 ( q ) , b 2 n + 1 ( q ) are the same until the terms in q 2 n - 2 and q 2 n , respectively. …
28.6.19 a - ( 2 n + 2 ) 2 - q 2 a - ( 2 n ) 2 - q 2 a - ( 2 n - 2 ) 2 - q 2 a - 2 2 = - q 2 ( 2 n + 4 ) 2 - a - q 2 ( 2 n + 6 ) 2 - a - , a = b 2 n + 2 ( q ) .
§28.6(ii) Functions ce n and se n
3: 28.2 Definitions and Basic Properties
Near q = 0 , a n ( q ) and b n ( q ) can be expanded in power series in q (see §28.6(i)); elsewhere they are determined by analytic continuation (see §28.7). …
4: 28.34 Methods of Computation
  • (a)

    Summation of the power series in §§28.6(i) and 28.15(i) when | q | is small.

  • (a)

    Summation of the power series in §§28.6(ii) and 28.15(ii) when | q | is small.

  • 5: 23.17 Elementary Properties
    §23.17(ii) Power and Laurent Series
    When | q | < 1
    23.17.4 λ ( τ ) = 16 q ( 1 - 8 q + 44 q 2 + ) ,
    In (23.17.5) for terms up to q 48 see Zuckerman (1939), and for terms up to q 100 see van Wijngaarden (1953). … with q 1 / 12 = e i π τ / 12 .
    6: 20.11 Generalizations and Analogs
    In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). …
    7: 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. Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. … Shanks (1955) applies such methods in several q -series problems; see Andrews et al. (1986).
    8: 16.5 Integral Representations and Integrals
    In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as z 0 in the sector | ph ( - z ) | ( p + 1 - q - δ ) π / 2 , where δ is an arbitrary small positive constant. …
    9: 16.8 Differential Equations
    For other values of the b j , series solutions in powers of z (possibly involving also ln z ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. … In this reference it is also explained that in general when q > 1 no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near z = 1 . Analytical continuation formulas for F q q + 1 ( a ; b ; z ) near z = 1 are given in Bühring (1987b) for the case q = 2 , and in Bühring (1992) for the general case. … If p q , then …Thus in the case p = q the regular singularities of the function on the left-hand side at α and coalesce into an irregular singularity at . …
    10: 3.10 Continued Fractions
    §3.10(ii) Relations to Power Series
    Stieltjes Fractions
    We say that it corresponds to the formal power seriesFor several special functions the S -fractions are known explicitly, but in any case the coefficients a n can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … We say that it is associated with the formal power series f ( z ) in (3.10.7) if the expansion of its n th convergent C n in ascending powers of z , agrees with (3.10.7) up to and including the term in z 2 n - 1 , n = 1 , 2 , 3 , . …