On application of Euler's differential method to a continued fraction depending on parameter (Q2018713)

Euler considered continued fractions of the form \[ K(t)=\frac{t}{1+}\frac{t+1}{2+}\frac{t+2}{3+}\frac{t+3}{4+}\dots \] For the function \(K(t+1)\) the authors give an integral representation \[ K(t+1)=\frac{\int_{0}^{1}(1-x)^{p-t}\frac{d^p}{dx^p}(x^{t+1}e^x)\,dx}{\int_{0}^{1}(1-x)^{p-t}\frac{d^p}{dx^p}(x^{t}e^x)\,dx},\qquad(p-1<t\leq p+1). \] In particular for integer \(p=1,2,\dots\) \[ K(t+1)=(p+1)\frac{\sum_{k=0}^{p-1}\frac{a_{p,k}}{p-k+1}}{\sum_{k=0}^{p-1}a_{p,k}}, \] where \[ a_{p,k}=\left({p\atop k}\right)\frac{1}{p-k+1)!}. \]