\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage[round]{natbib}
\let\cite\citep




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}










\begin{document}

\title{Laguerre Series: Functions in a Basis of Laguerre Polynomials}



\author[1]{Benedict Irwin}%
\affil[1]{University of Cambridge}%


\vspace{-1em}



  
  \date{May 24, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup








\sloppy




\section*{Abstract}
I show how to express functions with a known series expansion in terms of Laguerre polynomials.
\section*{Main}
We have the integral representation of the Laguerre polynomials, \begin{equation}
L_n(z)= \frac{e^z}{n!}\int_0^\infty e^{-t}t^n J_0(2\sqrt{ t z})\;dt,
\end{equation}
where $J_0(x)$ is a modified Bessel function, as given on the \href{http://functions.wolfram.com/Polynomials/LaguerreL/07/01/01/}{Wolfram Functions} website. If we have a function $f(x)$ that admits a series representation \begin{equation}
f(x)=\sum_{k=0}^\infty a_k x^k,
\end{equation}
with some coefficients $a_k$, then we can define a transform on $L[f]$ \begin{equation}
F(s)=L[f(x)]=\int_0^\infty K(x,s)f(x)\;dx.
\end{equation}
If we pick the kernel function to be \begin{equation}
K(x,s)= e^{-x}J_0(2\sqrt{xs}),
\end{equation}
then we have
\begin{equation}
F(s) = \sum_{k=0}^\infty a_k\int_0^\infty e^{-x}J_0(2\sqrt{xs})x^k \; dx = \sum_{k=0}^\infty \frac{a_k k!}{e^s}L_k(s),
\end{equation}
which is an expansion of the transformed function $F(s)$ in terms of Laguerre polynomials. As an example, if $f(x)=1$, then $a_k=[k=0]$, this gives \begin{equation}
\int_0^\infty e^{-x}J_0(2\sqrt{xs})\;dx = \sum_{k=0}^\infty \frac{[k=0]k!}{e^s}L_k(s) = \frac{L_0(s)}{e^s} = e^{-s}
\end{equation}
this leads to an infinitely recursive integral \begin{equation}
e^{-s} = \int_0^\infty \int_0^\infty \int_0^\infty \cdots \int_0^\infty e^{-x_n}J_0(2\sqrt{x_n x_{n-1}})\;dx_n   \cdots J_0(2\sqrt{x_3x_2})\;dx_3J_0(2\sqrt{x_2x_1})\;dx_2J_0(2\sqrt{x_1s})\;dx_1
\end{equation}
this should also lead to the set of results \begin{equation}
e^{-s} = \int_0^\infty e^{x_1}J_0(2\sqrt{x_1 s})\;dx_1
\end{equation}
\begin{equation}
e^{-s} = \int_0^\infty\int_0^\infty e^{x_2}J_0(2\sqrt{x_2x_1})J_0(2\sqrt{x_1 s})\;dx_1dx_2
\end{equation}
\begin{equation}

e^{-s} = \int_0^\infty\int_0^\infty\int_0^\infty e^{x_3}J_0(2\sqrt{x_3x_2})J_0(2\sqrt{x_2x_1})J_0(2\sqrt{x_1 s})\;dx_1dx_2dx_3
\end{equation}
and so on.

If we formally define a function \begin{equation}
f(x) = \sum_{k=0}^\infty \frac{x^k}{e k!k!} = \frac{1}{e} I_0(2\sqrt{x})
\end{equation}
we end up with a series expansion for $F(s)$, starting \begin{equation}
F(s) = 1 - 2s + \frac{7}{4}s^2 - \frac{17}{18}s^3 + \frac{209}{576}s^4 - \frac{773}{7200}s^5 + \cdots
\end{equation}

It seems the inverse kernel is given by \begin{equation}
K^{-1}(x,s) = \sum_{i=0}^\infty \sum_{j=0}^i \frac{(-1)^j s^j x^i}{(i-j)!j!j!} = \sum_{i=0}^\infty \frac{x^i}{i!}L_i(s)
\end{equation}
such that \begin{equation}
f(x) = \int_0^\infty K^{-1}(x,s)F(s)\;ds
\end{equation}

\selectlanguage{english}
\FloatBarrier
\end{document}

