Sie sind hier: ICP » R. Hilfer » Publikationen


[123.8.1] The H-function of order (m,n,p,q)\in\mathbb{N}^{4} and with parameters A_{i}\in\mathbb{R}_{+}(i=1,\ldots,p), B_{i}\in\mathbb{R}_{+}(i=1,\ldots,q), a_{i}\in\mathbb{C}(i=1,\ldots,p), and b_{i}\in\mathbb{C}(i=1,\ldots,q) is defined for z\in\mathbb{C},z\neq 0 by the contour integral [5, 32]

{(b_{1},B_{1}),\ldots,(b_{q},B_{q})}\end{array}\right.\right)=\frac{1}{2\pi i}\int _{{\mathcal{L}}}\eta(s)z^{{-s}}\;\mathrm{d}s (14)

where the integrand is

\eta(s)=\frac{\displaystyle\prod _{{i=1}}^{m}\Gamma(b_{i}+B_{i}s)\prod _{{i=1}}^{n}\Gamma(1-a_{i}-A_{i}s)}{\displaystyle\prod _{{i=n+1}}^{p}\Gamma(a_{i}+A_{i}s)\prod _{{i=m+1}}^{q}\Gamma(1-b_{i}-B_{i}s)}. (15)

[123.8.2] In (14) z^{{-s}}=\exp\{-s\log|z|-i\arg z\} and \arg z is not necessarily the principal value. [123.8.3] The integers m,n,p,q must satisfy

0\leq m\leq q,\qquad 0\leq n\leq p, (16)

and empty products are interpreted as being unity. [123.8.4] For the conditions on the other parameters and the path {\mathcal{L}} of integration the reader is referred to the literature [5] (see [13, p.120ff] for a brief summary).