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.

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    $\begingroup$ One of the top google search results: planetmath.org/encyclopedia/… $\endgroup$ – anon Dec 18 '11 at 21:25
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    $\begingroup$ Maybe it would help to note that there is a factor $\frac{1}{q!}$ missing in the Rodrigues's representation of Laguerre polynomials. $\endgroup$ – Sasha Dec 18 '11 at 21:33
  • $\begingroup$ @anon: Thank you very much for this link! $\endgroup$ – Sam Dec 18 '11 at 21:33
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    $\begingroup$ I think, I now have found a great one called Special Functions and Their Applications written by N. N. Lebedev. Any other suggestions are still very welcome though! $\endgroup$ – Sam Dec 18 '11 at 22:34
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    $\begingroup$ Yes, if you're trying to study orthogonal polynomials only, have a look at Chihara's and Szegő's books. For special functions in general, see the books by Temme, Carlson, and Andrews/Askey/Roy. $\endgroup$ – J. M. isn't a mathematician Dec 19 '11 at 2:32

Lebedev: Special functions and their applications

Bryon and Fuller: Mathematics of classical and quantum physics

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  • $\begingroup$ Welcome to Math.SE and thanks for your suggestions. I edited them to add links. $\endgroup$ – user53153 Jan 6 '13 at 2:30

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