[Math] Finding a generating function for the Laguerre polynomials

generating-functionsreference-requestspecial functions

I've started learning some quantum physics and one often encounters special functions (like Legendre polynomials, Laguerre polynomials, Bessel functions, …). Many calculations with these functions are immensely simplified (or maybe only made possible?) by making use of a generating function.

My physics book unfortunately does not give proofs that the generating functions do indeed generate the functions, and for example for the following I'm not able to do it myself:

It is stated that

$$U(\rho, s) = \frac{\exp[-\rho s/(1-s)]}{1-s} = \sum_{q=0}^\infty \frac{L_q(\rho)}{q!}s^q$$

is a generating function for the Laguerre polynomials $L_q(\rho)$, defined by

$$L_q(\rho) = e^\rho \frac{\mathrm d^q}{\mathrm d\rho^q}\left(\rho^q e^{-\rho}\right)$$

I have played around with it a bit, but wasn't able to show that $U(\rho, s)$ has the claimed series development around $s = 0$. Looking randomly through some books on special functions, I was not able to find a proof of this either. So my questions are:

  • What is a good book on special functions – one where I would find this stuff (the functions and relations mostly used in atomic physics, maybe)?
  • Can somebody show me how to prove this particular identity or give a reference?

The physics book I'm working on is Physics of Atoms and Molecules by Bransden and Joachain.

Related Question