Simplify the confluent hypergeometric function with parameters $(n,1,z)$

Question

Could you please provide a simple closed form expression for the confluent hyper-geometric function including the following (simple) parameters?

${}_1F_1(n,1,z)$

where n is positive integer and z is positive real.

Thank you for time and patience.

Answer 1

I suppose this is related to my answer here. Though I think it would be more useful for you do a minimum of work yourself (by e.g. going to this Wiki page), I feel certain responsibility. So here it is: for integer $n\geq0$

$$ _1F_1(n+1,1,z)=\frac{z^{-n}}{n!}\left(z\circ\frac{d}{dz}\circ z\right)^ne^z.$$

Answer 2

This confluent hypergeometric series is related to generalized Laguerre polynomials :

$$L_n^{(\alpha)}(x)=\frac{(\alpha + 1)_n}{n!} {}_1F_1(-n, \alpha + 1, x)$$

In particular,

$$L_n^{(0)}(x)=L_n(x)= {}_1F_1(-n, 1, x)$$

where $L_n(x)$ is the Laguerre polynomial, satisfying Laguerre's equation $xy''+(1-x)y'+ny=0$.

You can find more here about Laguerre polynomials.