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.