Regarding the Tricomi confluent hypergeometric function

Question

Is the following equation true for Tricomi confluent Hypergeometric function? $$\phi(1,0,ax)=1-ax\phi(1,1,ax)$$ here $\phi(.,.,.)$ is the Tricomi confluent hypergeometric function. Thanks in advance.

Answer 1

As usual examine carefully :)
Using DLMF 13.2.11, 13.6.6
$U(1,0,z)=z\cdot U\left(2,2,z\right)=e^{z}\cdot E_{2}\left(z\right)$
$z\cdot U(1,1,z)=z\cdot e^{z}E_{1}\left(z\right)$
So your question is
$E_{2}\left(z\right)+z\cdot E_{1}\left(z\right)=e^{-z}$
Which matches 8.19.12