# functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

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##### 1: 8.22 Mathematical Applications
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
2.11.11 $F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-% 1}}{1+t}\,\mathrm{d}t=\frac{\Gamma\left(n+p\right)}{2\pi}\frac{E_{n+p}\left(z% \right)}{z^{n+p-1}}.$
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 $E_{p}\left(z\right)$. … However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). …
2.11.20 $R_{n}^{(1)}(z)=(-1)^{n-1}ie^{(\mu_{2}-\mu_{1})\pi i}e^{\lambda_{2}z}z^{\mu_{2}% }\left(C_{1}\sum_{s=0}^{m-1}(-1)^{s}a_{s,2}\frac{F_{n+\mu_{2}-\mu_{1}-s}\left(% z\right)}{z^{s}}+R_{m,n}^{(1)}(z)\right),$
##### 3: 15.1 Special Notation
15.1.1 ${{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),$
##### 4: 32.14 Combinatorics
32.14.1 $\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$
where the distribution function $F(s)$ is defined here by The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). …
##### 5: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$, the Appell (two-variable hypergeometric) functions ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)$, ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)$, ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)$, and the Meijer $G$-function ${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$. Alternative notations are ${{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)$, ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z\right)$, and ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ for the generalized hypergeometric function, $F_{1}(\alpha,\beta,\beta^{\prime};\gamma;x,y)$, $F_{2}(\alpha,\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y)$, $F_{3}(\alpha,\alpha^{\prime},\beta,\beta^{\prime};\gamma;x,y)$, $F_{4}(\alpha,\beta;\gamma,\gamma^{\prime};x,y)$, for the Appell functions, and ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ for the Meijer $G$-function.
##### 7: 8.14 Integrals
8.14.5 $\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1+b;1/(1+s)\right),$ $\Re s>0$, $\Re\left(a+b\right)>0$,
8.14.6 $\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right),$ $\Re s>-1$, $\Re\left(a+b\right)>0$, $\Re a>0$.
For the hypergeometric function $F\left(a,b;c;z\right)$ see §15.2(i). …
##### 8: 16.2 Definition and Analytic Properties
###### §16.2(i) Generalized Hypergeometric Series
Equivalently, the function is denoted by ${{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)$ or ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$, and sometimes, for brevity, by ${{}_{p}F_{q}}\left(z\right)$. …
16.2.3 ${{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)$
16.2.5 ${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};$
When $p\leq q+1$ and $z$ is fixed and not a branch point, any branch of ${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ is an entire function of each of the parameters $a_{1},\dots,a_{p},b_{1},\dots,b_{q}$.
##### 9: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
###### §33.5(ii) $\eta=0$
For the functions $\mathsf{j}$, $\mathsf{y}$, $J$, $Y$ see §§10.47(ii), 10.2(ii). …
##### 10: 33.11 Asymptotic Expansions for Large $\rho$
###### §33.11 Asymptotic Expansions for Large $\rho$
$F_{\ell}\left(\eta,\rho\right)=g(\eta,\rho)\cos{\theta_{\ell}}+f(\eta,\rho)% \sin{\theta_{\ell}},$
$F_{\ell}'\left(\eta,\rho\right)=\widehat{g}(\eta,\rho)\cos{\theta_{\ell}}+% \widehat{f}(\eta,\rho)\sin{\theta_{\ell}},$
33.11.7 $g(\eta,\rho)\widehat{f}(\eta,\rho)-f(\eta,\rho)\widehat{g}(\eta,\rho)=1.$