# generalized hypergeometric series

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##### 1: 16.2 Definition and Analytic Properties
###### §16.2(i) GeneralizedHypergeometricSeries
Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$. … If none of the $a_{j}$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk $|z|<1$ the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … However, when one or more of the top parameters $a_{j}$ is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in $z$. … Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
##### 2: 34.6 Definition: $\mathit{9j}$ Symbol
The $\mathit{9j}$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
##### 3: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions
###### §16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions
Expansions of the form $\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)$ 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).
##### 4: 34.4 Definition: $\mathit{6j}$ Symbol
For alternative expressions for the $\mathit{6j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{4}F_{3}}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
##### 5: 34.2 Definition: $\mathit{3j}$ Symbol
For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
##### 6: Bibliography B
• W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
• W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
• W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.
• ##### 7: Bibliography P
• W. F. Perger, A. Bhalla, and M. Nardin (1993) A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.
• ##### 8: Bibliography W
• F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
##### 10: 16.11 Asymptotic Expansions
###### §16.11(i) Formal Series
16.11.6 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),$ $|\operatorname{ph}\left(-z\right)|\leq\pi$;
16.11.7 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).$
16.11.8 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),$
16.11.9 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),$