Some identities involving exponential functions and Stirling numbers and applications

Guo and Qi (2013) posed a problem asking to determine the coefficients ak,i−1ak,i−1 for 1≤i≤k1≤i≤k such that 1/(1−e−t)k=1+∑i=1kak,i−1(1/(et−1))(i−1). The authors answer this question alternatively by Faà di Bruno’s formula, unify the eight identities due to Guo and Qi to two identities involving two parameters, and apply them to obtain an explicit expression for the Apostol–Bernoulli numbers and the Fubini numbers, respectively.