# terminating

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##### 1: 8.22 Mathematical Applications
###### §8.22(i) Terminant Function
The so-called terminant function $F_{p}\left(z\right)$, defined by
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
##### 2: 2.11 Remainder Terms; Stokes Phenomenon
2.11.10 $E_{p}\left(z\right)=\frac{e^{-z}}{z}\sum_{s=0}^{n-1}(-1)^{s}\frac{{\left(p% \right)_{s}}}{z^{s}}+(-1)^{n}\frac{2\pi}{\Gamma\left(p\right)}z^{p-1}F_{n+p}% \left(z\right),$
2.11.11 $F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-% 1}}{1+t}\mathrm{d}t=\frac{\Gamma\left(n+p\right)}{2\pi}\frac{E_{n+p}\left(z% \right)}{z^{n+p-1}}.$
Owing to the factor $e^{-\rho}$, that is, $e^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_{p}\left(z\right)$. … In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). …
##### 3: 16.2 Definition and Analytic Properties
Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$. … 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 that if $-m$ is the value of the numerically largest $a_{j}$ that is a nonpositive integer, then the identity …
##### 4: 9.7 Asymptotic Expansions
9.7.20 $R_{n}(z)=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{G_{n-k}\left(2\zeta\right)% }{\zeta^{k}}+R_{m,n}(z),$
9.7.21 $S_{n}(z)=(-1)^{n-1}\sum_{k=0}^{m-1}(-1)^{k}v_{k}\frac{G_{n-k}\left(2\zeta% \right)}{\zeta^{k}}+S_{m,n}(z),$
9.7.22 $G_{p}\left(z\right)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z% \right).$
##### 5: 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. …
##### 6: 10.17 Asymptotic Expansions for Large Argument
If these expansions are terminated when $k=\ell-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\ell\geq\max(\nu-\tfrac{1}{2},1)$. …
10.17.16 $G_{p}\left(z\right)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z% \right),$
10.17.17 $R_{\ell}^{\pm}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m% -1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(\mp 2iz\right)+R_{m,\ell% }^{\pm}(\nu,z)\right),$
##### 7: 29.15 Fourier Series and Chebyshev Series
When $\nu=2n$, $m=0,1,\dots,n$, the Fourier series (29.6.1) terminates: … When $\nu=2n+1$, $m=0,1,\dots,n$, the Fourier series (29.6.16) terminates: … When $\nu=2n+1$, $m=0,1,\dots,n$, the Fourier series (29.6.31) terminates: … When $\nu=2n+1$, $m=0,1,\dots,n$, the Fourier series (29.6.8) terminates: … When $\nu=2n+2$, $m=0,1,\dots,n$, the Fourier series (29.6.46) terminates: …
##### 8: 4.6 Power Series
If $a=0,1,2,\dots$, then the series terminates and $z$ is unrestricted. …
##### 9: 16.4 Argument Unity
See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … when $\Re\left(2c-a-b\right)>-1$, or when the series terminates with $a=-n$. … when $\Re\left(b+c+d-a\right)<1$, or when the series terminates with $d=-n$. … when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …
##### 10: 10.40 Asymptotic Expansions for Large Argument
10.40.13 $R_{\ell}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m-1}% \frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(2z\right)+R_{m,\ell}(\nu,z)\right),$
where $G_{p}\left(z\right)$ is given by (10.17.16). …