# Questions tagged [laguerre-polynomials]

For questions about (associated) Laguerre polynomials, which arise in quantum physics.

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### How is the relation between the general Laguerre Differential equation and associated Laguerre differential equation deduced?

From Wikipedia, the Laguerre Differential Equation is defined as follows: \begin{align} x y'' + (v + 1 - x) y' + \lambda y = 0 \end{align} By definition, the solution of this differential equation is ...
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### Calculating the Laguerre function

In the Noncentral Chi Distribution Wikipedia page, the calculated Mean is: $${\sqrt{{\pi\over2}}L_{1/2}^{(k/2-1)} \left( {\small{-\lambda^2\over2}}\right)}$$ I am calculating the average distance ...
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### Proving that the Laguerre polynomials do indeed solve the differential equation

I am trying to show that the Laguerre differential equation, given in my homework problem as $xL''_n(x)+(1−x)L'_n(x)+ nL_n(x) = 0$, is indeed solved by the Laguerre polynomials in their closed sum ...
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### Sum over (squares of) Laguerre Polynomials

I'm looking for a closed form of the sum $$\sum_{n=0}^\infty \frac{n!}{(n+k)!} (L_n^k(x))^2 t^n,$$ where $L_n^k(x)$ are the Laguerre Polynomials. I have been looking for ...
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### A modification of the Laguerre product expansion

Given a product of Laguerre polynomials, $L_n(x) L_m(x)$, a particular question to ask is the expansion of this product in terms of the Laguerre polynomials $\{L_i(x)\}$ themselves. That is, we would ...
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### Is the sum of the first N Laguerre polynomials (with alternating signs) always positive?

I have noticed that the following simple sum of Laguerre polynomials (weighted with alternating signs) seems to be positive for any $N$ when $x>0$: $$\sum_{k=0}^{N}\;(-1)^{k}\;L_{k}(x)$$ More ...
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