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1.How to prove Rodrigues formula for Associated Laguerre Polynomial?

2.How to show they are orthonormal in the interval (0,infinity)? Also I want to find normalization constant?

3.How to prove recurrence relation and series expansion?

4.I am also interested to learn the proof of generating function,orthonormality,recurrence relation and series expansion for Associated Legandre Polynomials.

   As I am learning Quantum Mechanics , these are important to me too much.I can't find any resource that explains these topics.Please somebody helps me by putting the mathematical derivation or by creating link which contains the answer.
Deepnarayan Neogi
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1 Answers1

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How you prove it depends on what you're allowed to assume.

It could be that you're allowed to assume the Rodrigues formula and then show that $L_n^{\alpha}(x)$ solve the following differential equation:

$$xy'' + (\alpha + 1 - x)y' + ny = 0$$

This equation defines the associated Laguerre polynomials more than anything else, I'd think.

As for showing the recurrence relation you can start by calculating $L_0^{\alpha}(x)$ and $L_1^{\alpha}(x)$ and then use the Rodrigues relation and the differential equation to show the relation for $L_n^{\alpha}(x)$ in terms of $L_{n-1}^{\alpha}(x)$ and $L_{n-2}^{\alpha}(x)$.

The Wikipedia article on Laguerre polynomials (particularly the section with generalized Laguerre polynomials) should be a good start for you.

John
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