On the differences of the generalized factorials at an arbitrary point and their combinatorial applications
In the paper there are defined and systematically investigated the Gould- Hopper numbers G(n,k;r,s) useful in biology and reliability. First of all there are given the definitions and notations, e.g. \(G(m,n;r,s)=[1/n! \Delta^ n(rx)_ m]_{x=s/r}=[1/n! \Delta^ n(rx+s)_ m]_{x=0}\) otherwise we can define \(G(m,n;r,s)=1/n!\sum^{n}_{k=0}(-1)^{n- k}\left( \begin{matrix} n\\ k\end{matrix} \right)(rk+s)_ m.\) This expression leads to the combinatorial interpretations of the numbers G(m,n;r,s) when both r and s are positive or negative integers. Further are given the recurrence relation and generating function especially the exponential generating function, then the connection with other numbers (e.g. with Stirling numbers) and limiting expressions and the orthogonality relations and other properties. Finally there are given the combinatorial and statistical applications (e.g. sequential occupancy, coupons collector problem and convolution of complete and zero-truncated general binomial distributions).
scientific article; zbMATH DE number 3857120
scientific article