Several series containing gamma and polygamma functions (Q1392779)

The multiple integrals defined recursively by \[ \begin{aligned} I_{n+1}(z,w,x)&:=\int_x^1\frac{dy}{y-1}I_n(z,w,y), 0\leq x\leq 2,n=1,2,3,\ldots\\ I_1(z,w,x)&:=\int_x^1\frac{y^z-y^w}{y-1}dy \end{aligned} \] are studied. These functions are generalizations of the polylogarithmic functions \(Li_n(x):=\int_0^xLi_{n-1}(t)dt/t,Li_1(x):=-\log(1-x)\): \[ Li_{n+1}(x)=(-1)^{n+1}\left.\frac{d}{dz}I_n(z,w,1-x)\right|_{z=0}. \] Let \(F_n(z):= \frac{1}{n!}\Gamma(z+1)\frac{d^n}{dz^n} \left(\frac{1}{\Gamma(z+1)}\right)\). These functions are essentially polygamma functions. Then the following is proved: \[ I_n(z,w):=I_n(z,w,0)=\sum_{k=0}^nC_k(F_{n-k}(z)-F_{n-k}(w)), \] where \(C_k\) are determined recursively by \(C_k=-\sum_{i=0}^{k-1}C_iF_{k-i}(0),C_0=-1\). Using this, the author gives several new series involving gamma and polygamma functions.