Multiple Dirichlet series interpolating Bell numbers and Stirling numbers (Q983508)

Let \({\mathfrak P}_1(s)=\sum_{k=1}^\infty\frac{a_k}{k^s}\) be a Dirichlet series which is absolutely convergent for all \(s\in{\mathbb C}\), \[ {\mathfrak P}_r(q_1,2p_2,\dots,2p_{r-1},s):=\sum_{{\mathbf k}\in\Phi_r}\frac{a_{k_1}\text{sgn}(\sum_{j=1}^r k_j)}{k_1^{q_1} \prod_{d=2}^{r-1} (\sum_{j=1}^dk_j)^{2p_d} |\sum_{j=1}^r k_j|^s} \] for \(\Re(s)>1\), \(r\geq 3\), \(q_1\in {\mathbb C}\), \(p_2,\dots,p_{r-1}\in{\mathbb N}\), \[ \Phi_r:=\{{\mathbf k}=(k_j)\in{\mathbb N}\times({\mathbb Z}^*)^{r-1}\mid\sum_{j=1}^d k_j\neq 0, 2\leq d\leq r \}. \] The author proves that \[ {\mathfrak P}_r(q_1,2p_2,\dots,2p_{r-1},s)=(-1)^{r+1}{\mathfrak P}_1(q_1+2\sum_{j=2}^{r-1}p_j+s). \] As application one obtains multiple Dirichlet series interpolating Bell and Stirling numbers.