# expansions of solutions in series of

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##### 1: 31.11 Expansions in Series of Hypergeometric Functions
###### §31.11 ExpansionsinSeries of Hypergeometric Functions
$\mu=\gamma+\delta-2.$
###### §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.
##### 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 $\nu=\pm\widehat{\nu}+2n$, where $n\in\mathbb{Z}$. … The Fourier series of a Floquet solution …leads to a Floquet solution. … Near $q=0$, $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ 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), …
##### 8: 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).
##### 9: 2.2 Transcendental Equations
where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). …
##### 10: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions. In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …