  The paper is devoted to study the $H$-function defined by the Mellin-Barnes
integral
  $$H^{m,n}_{\thinspace p,q}(z)={\frac1{2\pi i}}\int_{\Lss}
\HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$
  where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a certain ratio of
products of Gamma functions with the argument $s$ and the contour $\LL$ is
specially chosen. The conditions for the existence of $H^{m,n}_{\thinspace
p,q}(z)$ are discussed and explicit power and power-logarithmic series
expansions of $H^{m,n}_{p,q}(z)$ near zero and infinity are given. The obtained
results define more precisely the known results.
