Integral representation of certain combinatorial recurrences (Q519921)

In [Aequationes Math. 80, No. 3, 291--318 (2010; Zbl 1226.05029)], the authors studied self-convolutive recurrence sequences \[ u_n=(\alpha_1n+\alpha_2)u_{n-1}+\alpha_3\sum_{j=1}^{n-1}u_ju_{n-j} \] with \(u_1=1\). They pointed out that such sequences, under mild assumptions, can be represented as moments of measures plus a finite term, i.e., \[ u_n=\int_0^\infty x^{n-1}\mu(x)dx-\sum_jr_j(-\zeta_j)^{n-1}, \] where \(r_j\), \(\zeta_j\) are some complex numbers. In the present study, the authors give concrete and interesting examples for such integral representations involving the exponential integral function. The examples come from permutation counting.