MacRobert's E-Function

$E(p; \alpha_r: \rho_s:x)\equiv {\Gamma(\alpha_{q+1})\over\Gamma(\rho_1-\alpha_1)\Gamma(\rho_2-\alpha_2)\cdots \Gamma(\rho_q-\alpha_q)}$

$\times \prod_{\mu=1}^q \int_0^\infty {\lambda_\mu}^{\rho_\mu-\alpha_\mu-1}(1+\l... ...nfty e^{-\lambda_{q+\nu}}{\lambda_{q+\nu}}^{\alpha_{q+\nu}-1}\,d\lambda_{q+\nu}$

$\times\int_0^\infty e^{-\lambda_p}{\lambda_p}^{\alpha_p-1}\left[{1+{\lambda_{q... ...p\over (1+\lambda_1)\cdots (1+\lambda_q)x}}\right]^{-\alpha_q+1}\!\!d\lambda_p,$

where $\Gamma(z)$ is the Gamma Function and other details are discussed by Gradshteyn and Ryzhik (1980).

See also Fox's H-Function, Meijer's G-Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 896-903 and 1071-1072, 1979.

$E(p; \alpha_r: \rho_s:x)\equiv {\Gamma(\alpha_{q+1})\over\Gamma(\rho_1-\alpha_1)\Gamma(\rho_2-\alpha_2)\cdots \Gamma(\rho_q-\alpha_q)}$
$\times \prod_{\mu=1}^q \int_0^\infty {\lambda_\mu}^{\rho_\mu-\alpha_\mu-1}(1+\l... ...nfty e^{-\lambda_{q+\nu}}{\lambda_{q+\nu}}^{\alpha_{q+\nu}-1}\,d\lambda_{q+\nu}$
$\times\int_0^\infty e^{-\lambda_p}{\lambda_p}^{\alpha_p-1}\left[{1+{\lambda_{q... ...p\over (1+\lambda_1)\cdots (1+\lambda_q)x}}\right]^{-\alpha_q+1}\!\!d\lambda_p,$