power series expansion of the Confluent Hypergeometric Function

Question

What is the power series expansion of the Confluent Hypergeometric Function of the Second Kind given by $U(a,b,x)$ ? what is the derivative of this function with respect to x.

Answer 1

Since: $$ U(a,b,x)=\frac{1}{\Gamma(a)}\int_{0}^{+\infty}e^{-xt}t^{a-1}(1+t)^{b-a-1}\,dt\tag{1}$$ we have:

$$\begin{eqnarray*} \frac{d}{dx}U(a,b,x) &=& -\frac{1}{\Gamma(a)}\int_{0}^{+\infty}e^{-xt}t^{a}(1+t)^{b-a-1}\,dt\\&=& -a\cdot U(a+1,b+1,x).\tag{2}\end{eqnarray*}$$