# Meijer G-function

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## 11—20 of 26 matching pages

##### 11: 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.
##### 14: 15.12 Asymptotic Approximations
where
15.12.13 $G_{0}(\pm\beta)=\left(2+e^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}\left(1+e^{\pm% \zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-e^{\pm\zeta}\right)^{-a+(\ifrac{1}% {2})}\sqrt{\frac{\beta}{e^{\zeta}-e^{-\zeta}}}.$
##### 15: 10.17 Asymptotic Expansions for Large Argument
10.17.17 $R_{\ell}^{\pm}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m% -1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(\mp 2iz\right)+R_{m,\ell% }^{\pm}(\nu,z)\right),$
##### 16: 5.17 Barnes’ $G$-Function (Double Gamma Function)
When $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)$, …
##### 18: 5.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$, the psi function (or digamma function) $\psi\left(z\right)$, the beta function $\mathrm{B}\left(a,b\right)$, and the $q$-gamma function $\Gamma_{q}\left(z\right)$. … Alternative notations for this function are: $\Pi(z-1)$ (Gauss) and $(z-1)!$. Alternative notations for the psi function are: $\Psi(z-1)$ (Gauss) Jahnke and Emde (1945); $\Psi(z)$ Davis (1933); $\mathsf{F}(z-1)$ Pairman (1919).
##### 19: Bibliography F
• J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.
• J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
• ##### 20: Bibliography M
• O. I. Marichev (1984) On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.
• C. S. Meijer (1946) On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).
• J. W. Meijer and N. H. G. Baken (1987) The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.