  We study the generalized Hankel transform of the family of sequences
satisfying the recurrence relation $a_{n+1} = \bigl(\alpha +
\frac{\beta}{n+\gamma}\bigr) a_n$. We apply the obtained formula to several
particular important sequences. Incidentally, we find a connection between some
well known formulas that had previously arisen in literature in dissimilar
settings. Additionally, given a non-zero sequence satisfying the above
recurrence, we evaluate the Hankel transform of the sequence of its
reciprocals.
