Logarithmic Power Series?

Question

Take $\lambda>0$ and $0<x<1$. What sort of insight does anybody have on the function $f(\lambda,x)=\lambda\sum_{k=0}^{\infty} \sum_{j=0}^k \frac{(-1)^j}{\lambda+j+1}\binom{k}{j}x^{\lambda+j+1}$. Is there any suggestions?

Answer 1

$$\frac{\partial f(\lambda,x)}{\partial x}=\lambda \sum\limits_{k=0}^{\infty}\sum\limits_{j=0}^k (-1)^j\binom{k}{j}x^{\lambda +j}=\lambda \sum\limits_{k=0}^{\infty}x^{\lambda}\sum\limits_{j=0}^k \binom{k}{j}(-x)^j=\lambda \sum\limits_{k=0}^{\infty}x^{\lambda}(1-x)^k=\frac{\lambda x^{\lambda}}{1-(1-x)}=\lambda x^{\lambda-1}$$ So $$f(\lambda,x)=\int\limits_0^x \lambda t^{\lambda-1}dt=x^{\lambda}$$