For positive integers
$t_{1},\ldots ,t_{k}$
, let
$\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$
(respectively
$p(n,t_{1},t_{2},\ldots ,t_{k})$
) be the number of partitions of
$n$
such that, if
$m$
is the smallest part, then each of
$m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$
appears as a part and the largest part is at most (respectively equal to)
$m+t_{1}+t_{2}+\cdots +t_{k}$
. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of
$p(n,t_{1},t_{2},\ldots ,t_{k})$
. We establish a
$q$
-series identity from which the formulae for the generating functions of
$\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$
and
$p(n,t_{1},t_{2},\ldots ,t_{k})$
can be obtained.