Derive Rodrigues’ formula for Laguerre polynomials
$$
L_n(x)=\frac{e^x}{n!}.\frac{d^n}{dx^n}(x^ne^{-x})
$$
The Rodrigues’ formula for Hermite polynomials can be obtained by taking $n^{th}$ order partial derivatives of its generatig function
$$
g(x,t)=\sum_{n=0}^\infty H_n(x)\frac{t^n}{n!}=1+tH_1(x)+\frac{t^2}{2!}H_2(x)+\cdots\cdots\cdots+\frac{t^n}{n!}H_n(x)+\cdots\cdots\cdots\\
\frac{\partial^n}{\partial t^n}\Big(e^{2xt-t^2}\Big)=H_n(x)+\frac{(n+1)n(n-1)\cdots2}{(n+1)!}tH_{n+1}(x)+\cdots\\
H_n(x)=\Bigg[\frac{\partial^n}{\partial t^n}\Big(e^{2xt-t^2}\Big)\Bigg]_{t=0}=e^{x^2}\Bigg[\frac{\partial^n}{\partial t^n}\Big(e^{-(x-t)^2}\Big)\Bigg]_{t=0}\\
=(-1)^ne^{x^2}\Bigg[\frac{\partial^n}{\partial x^n}\Big(e^{-(x-t)^2}\Big)\Bigg]_{t=0}=(-1)^ne^{x^2}\Bigg[\frac{\partial^n}{\partial x^n}\Big(e^{-x^2}\Big)\Bigg]
$$
Generating function for laguerre polynomials is $g(x,t)=\dfrac{e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^\infty L_n(x) t^n$
I do not think the same technique applies in the case of Laguerre polynomials. So how do I derive that for Laguerre polynomials ?
Best Answer
Note that the Taylor-Maclaurin formula cannot be used directly (as in the case of Hermite polynomials) as there is some $n$ dependency inside the $n^{th}$ derivative. How can we get around this ? ... Let use a coefficient extrator \begin{eqnarray*} x^n= [u^0]: \frac{u^{-n}}{1-xu}. \end{eqnarray*} We have \begin{eqnarray*} \sum_{n=0}^{\infty} L_n(x) t^n &=&[u^0]: e^x \sum_{n=0}^{\infty} \left(\frac{t}{u} \right)^n \frac{1}{n!} \frac{d^n}{dx^n} \frac{e^{-x}}{1-xu} \\ &=&[u^0]: e^x \frac{e^{-(x+t/u)}}{1-u(x+t/u)} \\ &=& \frac{1}{1-t} [u^0]: \frac{e^{-t/u}}{1-\frac{ux}{1-t}}. \\ \end{eqnarray*} Expand these two functions and observe that the central term is ... what we want \begin{eqnarray*} \left( \sum_{i=0}^{\infty} \frac{(-t/u)^i}{i!} \right) \left( \sum_{j=0}^{\infty} \left( \frac{ux}{1-t} \right)^j \right) = \cdots+ \sum_{k=0}^{\infty} \frac{1}{k!} \left( \frac{-xt}{1-t} \right)^k + \cdots \end{eqnarray*} Now reverse engineer all this ... and your result follows.