About the Project
NIST

power-series expansions in q

AdvancedHelp

(0.004 seconds)

1—10 of 16 matching pages

1: 28.15 Expansions for Small q
§28.15(i) Eigenvalues λ ν ( q )
2: 28.6 Expansions for Small q
§28.6(i) Eigenvalues
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 ) .
3: 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.

  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(ii)28.8(iv)).

  • (a)

    Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of q and z .

  • (c)

    Use of asymptotic expansions for large z or large q . See §§28.25 and 28.26.

  • 4: 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 .
    5: 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. …
    6: 3.10 Continued Fractions
    §3.10(ii) Relations to Power Series
    We say that it corresponds to the formal power series …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 n - 1 , n = 1 , 2 , 3 , . … For 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 , . …
    7: 19.5 Maclaurin and Related Expansions
    §19.5 Maclaurin and Related Expansions
    Series expansions of F ( ϕ , k ) and E ( ϕ , k ) are surveyed and improved in Van de Vel (1969), and the case of F ( ϕ , k ) is summarized in Gautschi (1975, §1.3.2). For series expansions of Π ( ϕ , α 2 , k ) when | α 2 | < 1 see Erdélyi et al. (1953b, §13.6(9)). …
    8: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    With 𝒞 μ ( j ) , c n ν ( q ) , A n m ( q ) , and B n m ( q ) as in §28.23, … In the case when ν is an integer, … The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the z -plane. For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
    9: 3.11 Approximation Techniques
    §3.11(ii) Chebyshev-Series Expansions
    In fact, (3.11.11) is the Fourier-series expansion of f ( cos θ ) ; compare (3.11.6) and §1.8(i). … For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). … be a formal power series. … When F has an explicit power-series expansion a possible choice of R is a Padé approximation to F . …
    10: 20.11 Generalizations and Analogs
    §20.11(ii) Ramanujan’s Theta Function and q -Series
    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). … As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). … For applications to rapidly convergent expansions for π see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). …