On a class of combinatorial sums involving generalized factorials
For the generalized falling factorial of \(t\) with increment \(\theta\) and order \(p\): \((t| \theta)_p = \prod_{j=0}^{p-1}(t-j \theta)\) \((p \geq 1)\), this paper gives a summation formula as follows: \[ \sum_{k=j}^n {k \choose j} (k + \lambda | \theta)_p = {n+1 \choose j} \sum_{r=0}^p \frac{(n+1-j)_{r+1}}{r+1+j} S(p,r;\theta,1,\lambda +j), \] where \(S(p,r;\theta,1,\lambda +j)\) are the generalized Stirling numbers.
scientific article; zbMATH DE number 5282459