Let \(p_1, \dots, p_t\) be distinct primes and \(\gcd(p_i -1, p_j-1) =2\) if \(i\neq j\). Let \(g\) be the common primitive root among \(v= p_1\cdots p_t\) and \(d := \mathrm{ord}_v(g)\). The Whiteman’s subgroup \(W^{(t)} = (g)\) is a subgroup of the multiplicative group \(\mathbb{Z}_{v}^*\) of order \(d\) (therefore the index is \(2^{t-1}\)). The reviewed paper studies the cyclotomic coset decomposition and the resulting Whiteman’s generalized cyclotomic numbers (definitions are given in Lemma 2.2 and Definition 2.3). The recurrence formulae of Whiteman’s generalized cyclotomic numbers with respect to \(p_1\cdots p_t\) are obtained (see Theorems 2.7, 2.8, 2.11). In particular, applying these recurrence formulae for \(t=3\) the authors obtain explicit generalized cyclotomic numbers with respect to \(p_1p_2p_3\).