Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

In this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

The total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.