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1: 31.11 Expansions in Series of Hypergeometric Functions
§31.11 Expansions in Series of Hypergeometric Functions
μ = γ + δ - 2 .
§31.11(v) Doubly-Infinite Series
Schmidt (1979) gives expansions of path-multiplicative solutions31.6) in terms of doubly-infinite series of hypergeometric functions.
2: Bibliography O
  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
  • 3: 28.5 Second Solutions fe n , ge n
    4: Howard S. Cohl
    Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
    5: 33.23 Methods of Computation
    Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. …
    6: 28.2 Definitions and Basic Properties
    The general solution of (28.2.16) is ν = ± ν ^ + 2 n , where n . … The Fourier series of a Floquet solution …leads to a Floquet solution. … 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). … For the connection with the basic solutions in §28.2(ii), …
    7: 18.40 Methods of Computation
    Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …
    8: 33.20 Expansions for Small | ϵ |
    §33.20(ii) Power-Series in ϵ for the Regular Solution
    9: 12.15 Generalized Parabolic Cylinder Functions
    This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).
    10: 2.2 Transcendental Equations
    where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …