This paper is devoted to the study of the H-function as defined by the
Mellin-Barnes integral
Hp,qm,n(z)=12πi∫ℒℋp,qm,n(s)z−sds,
where the function ℋp,qm,n(s) is a certain ratio of products of the Gamma-functions with the argument s and the contour ℒ specially chosen. The
conditions for the existence of Hp,qm,n(z) are discussed and explicit power
and power-logarithmic series expansions of Hp,qm,n(z) near zero and infinity
are given. The obtained results define more precisely the known results.