# expansions in series of hypergeometric functions

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##### 1: 31.11 Expansions in Series of Hypergeometric Functions

###### §31.11 Expansions in Series of Hypergeometric Functions

… ►###### §31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions.##### 2: 16.10 Expansions in Series of ${}_{p}F_{q}$ Functions

###### §16.10 Expansions in Series of ${}_{p}F_{q}$ Functions

… ►Expansions of the form ${\sum}_{n=1}^{\mathrm{\infty}}{(\pm 1)}^{n}{}_{p}F_{p+1}(\mathbf{a};\mathbf{b};-{n}^{2}{z}^{2})$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).##### 3: 14.32 Methods of Computation

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►In particular, for small or moderate values of the parameters $\mu $ and $\nu $ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
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##### 4: 13.24 Series

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►For expansions of arbitrary functions in series of ${M}_{\kappa ,\mu}\left(z\right)$
functions see Schäfke (1961b).
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##### 5: 12.20 Approximations

###### §12.20 Approximations

►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty}$ and $x=0$, respectively. As special cases of these results a Chebyshev-series expansion for $U(a,x)$ valid when $$ follows from (12.7.14), and Chebyshev-series expansions for $U(a,x)$ and $V(a,x)$ valid when $0\le x\le \lambda $ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …##### 6: 12.18 Methods of Computation

###### §12.18 Methods of Computation

►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …##### 7: Bibliography L

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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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##### 8: 16.25 Methods of Computation

###### §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. …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. …##### 9: Howard S. Cohl

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►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.
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##### 10: 6.20 Approximations

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for ${E}_{1}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $$ the scheme can be used in backward direction.