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\newtheorem*{requirements}{Requirements}

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\begin{document}

\mytitle[Associated Laguerre polynomials]{Associated Laguerre polynomials:\\
from the Rodrigues representation\\to the explicit formula}

\bigskip\noindent

\begin{objectives}
Calculate the coefficients of the
associated Laguerre polynomials $L_n^{(m)}$
starting from the Rodrigues representation:
\par\smallskip
\[
L_n^{(m)}(x)\eqdef \frac{1}{n!}\,x^{-m}\enumber^x\,
\frac{d^n}{dx^n}\Bigl(\enumber^{-x}\,x^{n+m}\Bigr).
\]
\end{objectives}

\begin{requirements}
General Leibniz rule, factorials.
\end{requirements}

\bigskip
\section*{Some products and factorials (short review)}

\medskip

\begin{exercise}
Simplify:
\par\smallskip
\[
\frac{7!}{4!}=\hsp{10em}\qquad\qquad\qquad
\frac{(n+2)!}{n!}=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Express the following product as a ratio of two factorials:
\par\smallskip
\[
\prod_{k=4}^{11} k = 4\cdot 5\cdots 11=\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $p,q\in\{1,2,\ldots\}$, $p<q$.
Express the following product as a ratio of two factorials:
\par\smallskip
\[
\prod_{k=p}^q k = p\cdots q=\hsp{20em}
\]
\end{exercise}

\clearpage
\section*{Derivatives of the exponential and monomial functions}

\medskip

\begin{exercise}
Calculate the first three derivatives of $x^m$ ($m\ge 2$):
\par
\begin{align*}
\frac{d}{dx}\bigl(x^m\bigr)&=\bigl(x^m\bigr)'=\hsp{15em}
\\[2ex]
\frac{d^2}{dx^2}\bigl(x^m\bigr)&=\bigl(x^m\bigr)''=\hsp{15em}
\\[2ex]
\frac{d^3}{dx^3}\bigl(x^m\bigr)&=\bigl(x^m\bigr)'''=\hsp{15em}
\end{align*}
\par\smallskip\noindent
In the last formula write the coefficient as a ratio of two factorials:
\par\smallskip
\[
\frac{d^3}{dx^3}\bigl(x^m\bigr)=\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate the $k$st derivative of $x^m$ ($k\le m$).
Write the coefficient as a ratio of two factorials.
\par\smallskip
\[
\frac{d^k}{dx^k}\bigl(x^m\bigr)=\bigl(x^m\bigr)^{(k)}=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate the $k$st derivative of $\enumber^{ax}$, where $a$ is a parameter:
\par\smallskip
\[
\frac{d^k}{dx^k}\bigl(\enumber^{ax}\bigr)=\hsp{15em}
\]
\end{exercise}

\bigskip
\clearpage
\section*{Derivatives of the product of the exponential function\\
by the monomial function}

\medskip

\begin{exercise}
Calculate the first three derivatives of the product $fg$
of two sufficiently smooth functions:
\par\smallskip
\begin{align*}
(fg)'&=\hsp{25em}
\\[2ex]
(fg)''&=\hsp{25em}
\\[4ex]
(fg)'''&=\hsp{25em}
\end{align*}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Write the general Leibniz rule:
\par\bigskip
\[
(fg)^{(n)}=\sum_{k=\ }\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Expand the following derivative.
Write the sum in such an order that the powers of $x$ form an increasing sequence.
Then factorize the exponential function and the maximal possible power of the monomial:
\par\smallskip
\[
\Bigl(\enumber^{ax}\,x^p\Bigr)' =\,\ubrace{3em}\,\enumber^{ax}\,x^{p-1}\,+\,\ubrace{3em}\,\enumber^{ax}\,x^p
= \enumber^{ax}\,\underbrace{\hsp{3em}}_{?}\,\Bigl(\hsp{8em}\Bigr).
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:derivative_exp_monomial}
Expand the following derivative using the general Leibniz rule.
Write the sum in such an order that the powers of $x$ form an increasing sequence.
Then factorize the exponential function and the maximal possible power of the monomial:
\par
\begin{align*}
\frac{d^n}{dx^n}\Bigl(\enumber^{ax}\,x^p\Bigr)
&=\sum_{k=\ }\hsp{20em}
\\[4ex]
&=\enumber^{ax}\,\underbrace{\hsp{3em}}_{x^?}\,\sum_{k=\ }\,\frac{\hsp{6em}}{\hsp{6em}}\;x^k.
\end{align*}
\end{exercise}

\bigskip
\clearpage
\section*{Definition of the associated Laguerre polynomials}

\medskip

\begin{defn}[associated Laguerre polynomials]
For all $n,m\in\{0,1,2,\ldots\}$, define the \emph{associated Laguerre polynomial} $L_n^{(m)}$ by the formula:
\[
L_n^{(m)}(x)\eqdef \frac{1}{n!}\,x^{-m}\enumber^x\,
\frac{d^n}{dx^n}\Bigl(\enumber^{-x}\,x^{n+m}\Bigr).
\]
\end{defn}

\bigskip

\begin{note}
Formulas defining some polynomials in this manner
(through $n$st derivatives of some products)
are called \emph{Rodrigues representation} or \emph{Rodrigues formulas}.
\end{note}

\bigskip

\begin{exercise}
Expand the following derivative using the result of the Exercise \ref{exer:derivative_exp_monomial}.
Factorize from the sum the exponential function and the maximal possible power of the monomial.
\par\medskip
\[
\frac{d^n}{dx^n}\Bigl(\enumber^{-x}\,x^{n+m}\Bigr)
=\enumber^{-x}\hsp{3em}\sum\hsp{10em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Write the associated Laguerre polynomial $L_n^{(m)}$ in the explicit form $\displaystyle\sum_{k=0}^?\ ?\ x^k$:
\par\medskip
\[
L_n^{(\alpha)}(x)=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Express the following derivative through the associated Laguerre polynomial ($n\le p$):
\par\smallskip
\[
\frac{d^n}{dx^n}\Bigl(\enumber^{-x}\,x^p\Bigr)=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Let $n<p$ and $\displaystyle h(x)\eqdef\frac{d^n}{dx^n}\Bigl(\enumber^{-x}\,x^p\Bigr)$.
Calculate:
\par\medskip
\[
h(0)=\ubrace{3em}\qquad\qquad\qquad\qquad \lim_{x\to+\infty}h(x)=\ubrace{3em}
\]
\end{exercise}

\bigskip

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