For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $ν_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.