On Hermite-Bell inverse polynomials (Q1059200)

Bell introduced a set of polynomials by \[ \exp g(z)(d^ n/dz^ n)\exp [-g(z)]=Y_ n(g:z)\quad where\quad g(z)=\sum^{\infty}_{n=1}a_ nz^ n. \] In the present paper a related set of polynomials is considered for \(g(z)=pz^{-k}\), where p is a constant and K is a positive integer. The polynomials \(Y_ n(pz^{-k};z)\) are polynomials in \(z^{-1}\), and the author obtains an explicit formula and a differential equation for these polynomials.