Fox's H-Function

A very general function defined by

$$\begin{eqnarray*} H(z)&=& {\bf H}_{p,q}^{m,n}\left[{z \Big\vert \begin{array}{c... ..._j+\beta_j s)\prod_{j=n+1}^{qp} \Gamma(a_j-\alpha_j s)} z^s\,ds, \end{eqnarray*}$$

where $0\leq m\leq q$, $0\leq n\leq p$, $\alpha_j,\beta_j>0$, and $a_j,b_j$ are Complex Numbers such that the pole of $\Gamma(b_j-\beta_j s)$ for $j=1$, 2, ..., $m$ coincides with any Pole of $\Gamma(1-a_j+\alpha_j s)$ for $j=1$, 2, ..., $n$. In addition $C$, is a Contour in the complex $s$-plane from $\omega-i\infty$ to $\omega+i\infty$ such that $(b_j+k)/\beta_j$ and $(a_j-1-k)/\alpha_j$ lie to the right and left of $C$, respectively.

See also MacRobert's E-Function, Meijer's G-Function

References

Carter, B. D. and Springer, M. D. ``The Distribution of Products, Quotients, and Powers of Independent $H$-Functions.'' SIAM J. Appl. Math. 33, 542-558, 1977.

Fox, C. ``The $G$ and $H$-Functions as Symmetrical Fourier Kernels.'' Trans. Amer. Math. Soc. 98, 395-429, 1961.