A unified approach to generalized Stirling numbers
Set \((z| \alpha)_n =z(z-\alpha) \cdots (z-n \alpha+ \alpha)\) and \((z | \alpha)_0=1\). For complex parameters \((\alpha, \beta,r) \neq(0,0,0)\), the paper studies the inverse relations \[ (t|\alpha)_n=\sum^n_{k=0} S^1(n,k) (t-r | \beta)_k \quad \text{and} \quad (t | \beta)_n =\sum^n_{k=0} S^2(n,k) (t+r | \alpha)_k. \] The resulting generalized Stirling numbers include binomial coefficients, Lah numbers, signless Stirling numbers, and many other generalized Stirling numbers. Generating functions and Dobinski-type formulae are also given.
scientific article; zbMATH DE number 1221690
scientific article