  The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied
for integer n>=-1 and positive parameter q. It is proved that the Hankel
transform is given by uniformly and absolutely convergent series in reciprocal
powers of q, provided special conditions on the function f(x) and its
derivatives are imposed. It is necessary to underline that similar formulas
obtained previously are in fact asymptotic expansions only valid when q tends
to infinity. If one of the conditions is violated, our series become asymptotic
series. The validity of the formulas is illustrated by a number of examples.
