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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.

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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 $n\geq0$

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

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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.

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  • $\begingroup$ This is not quite what is needed, though very much related. Applying Kummer transformation $_1F_1(a,b,z)=e^{z}\,_1F_1(b-a,b,-z)$, one finds that $_1F_1(n+1,1,z)=e^z\,L_n(-z)=e^z\sum_{k=0}^n\frac{n!\,z^k}{(k!)^2(n-k)!}$, which is the same as below. $\endgroup$ – Start wearing purple Apr 19 '13 at 14:57

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