The Apostol-Bernoulli polynomials \(B_n(u, z)\) were introduced by T. M. Apostol [Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] and are a natural generalisation of the Bernoulli polynomials. They have the exponential generating function
\[ \left(\frac{x}{ze^{x}-1}\right)e^{u x}=\sum_{n=0}^{\infty}B_{n}(u, z) \frac{x^n}{n!}, \]
where \(B_n(0, z)\) are the Apostol-Bernoulli numbers, \(B_n(u, 1)\) are the Bernoulli polynomials \(B_n(u)\), and \(B_n(0, 1)\) are the Bernoulli numbers \(B_n\).
In this present work the authors derive two new closed form representations for \(B_{n}(u, z)\). The first involves sums of products of Stirling numbers of the second kind (see Theorem 1.1), and the second as a determinant (see Theorem 1.4). In particular the Stirling number representation theorem states that for \(z\ne 1\) and \(n\in \mathbb{N}\), the Apostol-Bernoulli polynomials \(B_n(u, z)\) may be expressed as
\[ B_n(u, z) = n \sum_{k=0}^{n-1}\frac{(-1)^k k!}{(z-1)^{k+1}} \sum_{r+s=k}\,\, \sum_{l+m=n-1}(-1)^{s+m}\ \binom{n-1}{\ell} \]
\[ \times z^{r}(1-u)^{\ell}u^{m}S(\ell,\ r)S(m,\ s). \]
It follows that for \(n\in \mathbb{N}\), the Apostol-Bernoulli numbers \(B_n(z)\) can be represented as
\[ B_n(z)=n \sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{(z-1)^{k+1}}z^{k}S(n-1,\ k). \]
The proof of Theorem 1.1 involves Bell polynomials and the Faà di Bruno formula for the higher order derivatives of composite functions.