# expansions in series of hypergeometric functions

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## 11—20 of 62 matching pages

##### 11: 6.20 Approximations
• Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function13.2(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 $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

• ##### 12: 33.6 Power-Series Expansions in $\rho$
###### §33.6 Power-SeriesExpansionsin$\rho$
or in terms of the hypergeometric function (§§15.1, 15.2(i)),
33.6.4 $A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}% }\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).$
The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. Corresponding expansions for ${H^{\pm}_{\ell}}'\left(\eta,\rho\right)$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).
##### 13: 13.13 Addition and Multiplication Theorems
13.13.12 $e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),$ $|y|<|x|$.
##### 14: 35.10 Methods of Computation
###### §35.10 Methods of Computation
For small values of $\|\mathbf{T}\|$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. … See Yan (1992) for the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8). Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.
##### 15: 13.26 Addition and Multiplication Theorems
13.26.12 $e^{\frac{1}{2}y}\left(\frac{x}{x+y}\right)^{\kappa}\sum_{n=0}^{\infty}\frac{1}% {n!}\left(\frac{-y}{x+y}\right)^{n}W_{\kappa+n,\mu}\left(x\right),$ $\Re\left(y/x\right)>-\frac{1}{2}$.
##### 16: 33.23 Methods of Computation
###### §33.23(i) Methods for the Confluent HypergeometricFunctions
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. … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions3.11(iv)) for the analytic continuations of Coulomb functions. Noble (2004) obtains double-precision accuracy for $W_{-\eta,\mu}\left(2\rho\right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …
##### 17: 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 $\left|q\right|<1$, 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 functions22.2), and Weierstrass elliptic functions23.6(ii)) can be expanded in $q$-series via (20.9.1). … For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …
##### 18: 13.11 Series
###### §13.11 Series
For $z\in\mathbb{C}$, …
$(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\dots$.
For additional expansions combine (13.14.4), (13.14.5), and §13.24. For other series expansions see Tricomi (1954, §1.8), Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), López and Temme (2010a) and López and Pérez Sinusía (2014). …
##### 19: Bibliography O
• A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
• A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
• A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
• F. W. J. Olver (1991b) Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
• 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.
• ##### 20: 15.19 Methods of Computation
###### §15.19 Methods of Computation
For $z\in\mathbb{R}$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. … The representation (15.6.1) can be used to compute the hypergeometric function in the sector $|\operatorname{ph}\left(1-z\right)|<\pi$. … In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …