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\theoremstyle{definition}
\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

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\begin{document}

\mytitle{Another special approximate unit}

\begin{objectives}
To study some basic properties of the functions $\phi_n$ and their Laplace transforms $\psi_n$.
Here $\phi_n$ is defined on $\PositiveRealNumbers$ by\par\smallskip
\[
\phi_n(v)\eqdef\frac{1}{n!\,(n-2)!}\,\frac{d^n}{dv^n}\left(\enumber^{-v}\,v^{2n-1}\right).
\]
\end{objectives}

\begin{requirements}
Gamma function and its basic properties,
beta function and its basic properties,
expression of the beta function through the gamma function,
basic integration tecnics (change of variables and integration by parts),
general Leibniz rule,
Laplace transform.
\end{requirements}

\bigskip

\section*{Definition of $\boldsymbol{\phi_n}$}

\bigskip
Denote $(0,+\infty)$ by $\PositiveRealNumbers$.

\begin{defn}
For all $n\in\{1,2,\ldots\}$ define the function $\phi_n$ on $\PositiveRealNumbers$ by\par\smallskip
\begin{equation} \label{eq:phi_def}
\phi_n(v)\eqdef\frac{1}{n!\,(n-2)!}\,\frac{d^n}{dv^n}\left(\enumber^{-v}\,v^{2n-1}\right).
\end{equation}
\end{defn}

\bigskip

\begin{exercise}
Calculate the first derivative of $\enumber^{-v}\,v^n$:\par\smallskip
\[
\frac{d}{dv}\left(\enumber^{-v}\,v^n\right)=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate $\phi_2(v)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate $\phi_3(v)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\clearpage
\begin{exercise}
Recall the Binomial Formula:\par\smallskip
\[
(a+b)^n=\sum\hsp{7em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Calculate the first three derivatives of the product $fg$.
Here $f$ and $g$ are functions of one variable denoted by $v$.\par\smallskip
\begin{align*}
\frac{d}{dv}(fg)&=f' g + f g',\\[2ex]
\frac{d^2}{dv^2}(fg)&= \\[2ex]
\frac{d^3}{dv^3}(fg)&=\hsp{20em}
\end{align*}
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Write the formula (\emph{general Leibniz rule}) for the $k$-st derivative of a product:\par\smallskip
\[
\frac{d^k}{dv^k}(fg)=\sum\hsp{10em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\begin{exercise}
Calculate the $k$-st derivative of a power $v^n$, $k\le n$.
Use factorials or the gamma function to express the coefficient.\par\smallskip
\[
\frac{d^k}{dv^k}\bigl(v^n\bigr)=\hsp{5em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\clearpage
\begin{exercise}
Recall the definition \eqref{eq:phi_def} of the function $\phi_n$:\par\smallskip
\[
\phi_n(v)=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip

\begin{exercise} \label{exer:phi_n_explicit_form}
Express the function $\phi_n$ as a certain sum (free of derivatives):\par\smallskip
\begin{equation} \label{eq:phi_n_explicit_form}
\phi_n(v)=\hsp{5em}\sum\hsp{25em}
\end{equation}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
The previous exercise permit us to make some conclusions
the form of the function $\phi_n$.
The function $\phi_n$ can be written as a product\par\smallskip
\[
\ubrace{3em}Q_n(v)
\]
where $Q_n$ is a polynomial,\par\smallskip
\[
\deg(Q_n)=\ubrace{4em},
\]
and the coefficients of $x^k$ for $k<\ubrace{4em}$ are equal to zero.
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Formally we define the functions $\phi_n$ only on $\PositiveRealNumbers=(0,+\infty)$,
but they can be extended continuously onto $(-\infty,+\infty]$.
In particular, we are interested in the (limit) values of $\phi_n$ at the points $0$ and $+\infty$:
\par\smallskip
\begin{align*}
\lim_{v\to0^+}\phi_n(v) &= \hsp{20em}
\\[5ex]
\lim_{v\to+\infty}\phi_n(v) &= \hsp{20em}
\end{align*}
\end{exercise}

\clearpage
\subsection*{Functions $\boldsymbol{\phi_n}$ and associated Laguerre polynomials}

\bigskip

\begin{exercise}
Find the definition for the \textbf{associated Laguerre polynomials}
(= generalized Laguerre polynomials = Sonine polynomials)
in the following form (the Rodrigues representation):\par\bigskip
\begin{equation} \label{eq:associated_Laguerre_in_Rodrigues_representation}
L_m^k(v)=\hsp{5em}\frac{d^m}{dv^m}\biggl(\enumber^{-v}\hsp{4em}\biggr).
\end{equation}
Find also the explicit formula for the associated Laguerre polynomials:\par\bigskip
\begin{equation} \label{eq:associated_Laguerre_in_explicit_form}
L_m^k(v)=\sum\hsp{10em}
\end{equation}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Comparing the definition \eqref{eq:phi_def} of $\phi_n$
with \eqref{eq:associated_Laguerre_in_Rodrigues_representation}
express $\phi_n$ through some associated Laguerre polynomial.\par\bigskip
\begin{equation} \label{eq:phi_n_through_associated_Laguerre}
\phi_n(v)=\hsp{10em}
\end{equation}
\end{exercise}

\bigskip\bigskip

\begin{exercise}
Substitute \eqref{eq:associated_Laguerre_in_explicit_form}
into \eqref{eq:phi_n_through_associated_Laguerre}
and compare the obtained result with the formula \eqref{eq:phi_n_explicit_form}
from Exercise \ref{exer:phi_n_explicit_form}.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip


\clearpage
\section*{Some basic properties of the gamma function (review)}

\medskip

\begin{exercise}
Recall the definition of the gamma function:
\[
\Gamma(x)\eqdef\int_0^{+\infty}\ubrace{8em}\,dt.
\]
\end{exercise}

\medskip

\begin{exercise}
Integrating by parts express $\Gamma(x+1)$ through $\Gamma(x)$:\par\smallskip
\[
\Gamma(x+1)=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Compute $\Gamma(1)$:\par\smallskip
\[
\Gamma(1)=\int_0^{+\infty}\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Compute $\Gamma(2)$, $\Gamma(3)$, $\Gamma(4)$, $\Gamma(5)$:\par\smallskip
\begin{align*}
\Gamma(2)&=\ubrace{2em}\Gamma(1)=
&
\Gamma(3)&=\ubrace{2em}\Gamma(2)=
\\[3ex]
\Gamma(4)&=
&
\Gamma(5)&=
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Express $\Gamma(n)$ through the factorial function for $n\in\{1,2,3,\ldots\}$.\par\smallskip
\[
\Gamma(n)=\hsp{5em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a>0$ and $p>0$.
Express the following integral through the gamma function
(make a suitable change of variables):\par\smallskip
\[
\int_0^{+\infty}x^p \enumber^{-ax}\,dx=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\section*{Some basic propierties of the beta function (review)}

\begin{exercise}
Recall the definition of the beta function:\par\smallskip
\[
\Beta(x,y)\eqdef\int_0^1\ubrace{10em}\,du.
\]
\end{exercise}

\bigskip

\begin{exercise}
Express the following integral through the beta function:\par\smallskip
\[
\int_0^1 u^\alpha (1-u)^\beta\,du = \Beta(\,\ubrace{6em}\,).
\]
\end{exercise}

\bigskip

\begin{exercise}
Recall the formula that expresses the beta function through the gamma function:\par\smallskip
\[
\Beta(x,y)=\frac{\hsp{10em}}{\hsp{10em}}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $p,q\in\{1,2,3,\ldots\}$.
Using the formula from the previous exercise
express $\Beta(p,q)$ through some factorials.
\par\medskip
\[
\Beta(p,q)=\hsp{30em}
\]
\end{exercise}

\clearpage

\begin{exercise}
Prove that the function $\displaystyle t\mapsto\frac{t}{t+1}$
is strictly increasing on $\PositiveRealNumbers$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate the limits:\par\smallskip
\[
\lim_{t\to0}\frac{t}{t+1}=\hsp{8em}
\qquad\qquad
\lim_{t\to+\infty}\frac{t}{t+1}=\hsp{8em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Find the range of the function defined on $\PositiveRealNumbers$ by $t\mapsto\frac{t}{t+1}$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\displaystyle u=\frac{t}{t+1}$ where $t>0$. Express $t$ through $u$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Making a suitable change of variables express the following integral through the beta function:\par\smallskip
\[
\int_0^{+\infty}\frac{t^a\,dt}{(1+t)^b}=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip


\clearpage
\section*{$\boldsymbol{\psi_n\eqdef}$ the Laplace transform of $\boldsymbol{\phi_n}$}

\medskip\noindent
Recall the definition of $\phi_n$:
\[
\phi_n(v)\eqdef\frac{1}{n!\,(n-2)!}\,\frac{d^n}{dv^n}\left(\enumber^{-v}\,v^{2n-1}\right).
\]

\begin{defn}
For each $n\in\{1,2,\ldots\}$ define the function $\psi_n\colon\PositiveRealNumbers\to\RealNumbers$
as the Laplace transform of the function $\phi_n$:
\begin{equation} \label{eq:phi_def_as_Laplace_transform}
\psi_n(t)\eqdef\int_0^{+\infty}\phi_n(v)\,\enumber^{-vt}\,dv.
\end{equation}
\end{defn}

\bigskip

\begin{exercise}
Calculate $\psi_1$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate $\psi_2$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Calculate $\psi_3$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Using the definition \eqref{eq:phi_def} of $\phi_n$ calculate $\psi_n$.
The coefficient in the answer can be written through the beta function.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip



\clearpage
\section*{Some properties of $\boldsymbol{\psi_n}$}

\begin{exercise}
Let $n\in\{2,3,\ldots\}$.
Calculate the integral of $\psi_n$ on $\PositiveRealNumbers$:
\[
\int_0^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip

\begin{exercise}
To verify the result of the previous exercise, calculate the following integral:
\[
\int_0^{+\infty}\psi_2(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Prove that
\[
\lim_{n\to\infty}\sup_{0<t\le\enumber^{-\delta}}\psi_n(t)=0.
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:\par\medskip
\[
\lim_{n\to\infty}\int_0^{\enumber^{-\delta}}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip\bigskip\bigskip

\begin{exercise}
Let $\delta>0$.
Make the change of variables $\displaystyle s=\frac{1}{t}$ in the following integral:\par\medskip
\[
\int_{\enumber^{\delta}}^{+\infty}\psi_n(t)\,dt=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $\delta>0$.
Calculate the limit:\par\medskip
\[
\lim_{n\to\infty}\int_{\left(0,\enumber^{-\delta}\right)\cup\left(\enumber^{\delta},+\infty\right)}\psi_n(t)\,dt=\hsp{20em}
\]
\end{exercise}

\label{endlabel}
\end{document}





\bigskip

\begin{lem} \label{lem:gamma_Phi}
For all $n\in\{1,2,\ldots\}$, $x>0$, $y>0$,
\[
2x\,\int_0^{+\infty}\frac{\phi_n(2vy)}{y}\,\enumber^{-2xv}\,dv
=\frac{x}{y^2}\,\psi_n\left(\frac{x}{y}\right)
=\frac{1}{\Beta(n,n)}\,\frac{x^n\,y^{n-1}}{(x+y)^{2n}}.
\]
\end{lem}

