Some identities involving derangement polynomials and numbers and moments of gamma random variables (Q2221332)

The derangement number \(D_n\) is the number of fixed point free permutations on a set with \(n\) elements. It is given by the expression \(D_n=n! \sum_{k=0}^n (-1)^k/k!.\) The exponential generating function for \(D_n\) equals \[ \sum_{n=0}^\infty D_n \frac{t^n}{n!} = \frac{e^{-t}}{1-t}. \] This allows to consider three families of derangement polynomials: the ordinary derangement polynomials \(D_n(x)\), the bivariate sine-derangement polynomials \(D_n^{s}(x,y)\) and the bivariate cosine-derangement polynomials \(D_n^{c}(x,y)\), respectively, being defined by \[ \sum_{n=0}^\infty D_n(x) \frac{t^n}{n!} = \frac{e^{-t}}{1-t} e^{xt}, \] \[ \sum_{n=0}^\infty D_n^{s}(x,y) \frac{t^n}{n!} = \frac{e^{-t}}{1-t} e^{xt}\sin (y t), \] and \[ \sum_{n=0}^\infty D_n^{c}(x,y) \frac{t^n}{n!} = \frac{e^{-t}}{1-t} e^{xt}\cos (y t). \] The authors study these polynomial families and their applications to moments of some variants of gamma random variables.