  Assumed that the parameters of a generalized hypergeometric function depend
linearly on a small variable $\varepsilon$, the successive derivatives of the
function with respect to that small variable are evaluated at $\varepsilon=0$
to obtain the coefficients of the $\varepsilon$-expansion of the function. The
procedure, quite naive, benefits from simple explicit expressions of the
derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols
with respect to their argument. The algorithm may be used algebraically,
irrespective of the values of the parameters. It reproduces the exact results
obtained by other authors in cases of especially simple parameters. Implemented
numerically, the procedure improves considerably the numerical expansions given
by other methods.
