\documentclass{amsart}
\title{Homework 1}
\author{}

\usepackage{amsmath,amssymb}
\newcommand{\ds}{\displaystyle}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}

\begin{document}
\maketitle

\problem{1} Let $\sum_{k=0}^{\infty} a_kz^k \in \C[[z]]$ be a power
series with radius of convergence $1$. Let $\{y_n\}_{n\geq 0}$
be defined by:
\[
y_0=1,\qquad \text{and} \qquad
y_n = \frac{1}{n} \sum_{j=0}^{n-1} a_jy_{n-1-j}, \forall n\geq 1.
\]
Prove that the radius of convergence of 
$\sum_{k=0}^{\infty} y_k z^k$ is at least $1$.\\

\problem{2} Recall the definition of the hypergeometric series:
\[
F(a,b;c;z) = \sum_{n=0}^{\infty} \frac{z^n}{n!} \frac{(a)_n (b)_n}{(c)_n},
\]
where we use the notation $(p)_n = p(p+1)\cdots (p+n-1)$ if $n\geq 1$,
and $(p)_0=1$.
\begin{itemize}
\item[(i)] Prove that $\ds \frac{d}{dz} F(a,b;c;z) = 
\frac{ab}{c} F(a+1,b+1;c+1;z)$.\\

\item[(ii)] Prove that $F(a,b;c;z)$ is a solution of the following
second order linear ODE (called the hypergeometric equation).
\[
z(1-z)u'' + (c-(a+b+1)z) u' -ab u = 0.
\]

\item[(iii)] Prove that $(1-z)^{c-a-b} F(c-a,c-b;c;z)$ also solves
the hypergeometric equation, and takes value $1$ at $z=0$. Use this
to conclude that $F(a,b;c;z) = (1-z)^{c-a-b} F(c-a,c-b;c;z)$.
\end{itemize}

\problem{3} Verify the solution given in Lecture 2, \S 3, page 5.\\
{\bf (Very Optional)} Compute the matrix $K$ relating the solutions
near $z=0$ and $z=1$ (see Lecture 1, page 8) for the differential
equation given in Lecture 2, \S 3.\\

\problem{4} Let $z^{-1}\C[[z^{-1}]]$ be the ring of formal series
in $z^{-1}$ with zero constant term. Consider the (formal) Borel
transform: $\ds z^{-1}\C[[z^{-1}]] \to \C[[p]]$ given by:
\[
\mathcal{B} : \sum_{n\geq 0} c_n z^{-n-1} \mapsto \sum_{n\geq 0} c_n \frac{p^n}{n!}
\]
which we will denote by $F(z) \mapsto \mathcal{B}(F)(p)$.
\begin{enumerate}
\item Prove that $\mathcal{B}(-\partial_z F)(p) = p\mathcal{B}(F)(p)$.\\

\item Let $c\in \C$ and $F(z)\in z^{-1}\C[[z^{-1}]]$. Consider the
shift operator $(T_cF)(z) := F(z-c)$. Prove that
$\mathcal{B}(T_cF)(p) = e^{cp}\mathcal{B}(F)(p)$.\\

\item Compute $\mathcal{B}(F)$ for the following $F(z)$:
\[
F(z) = \frac{1}{(z-c)^\ell}, \ell \in \Z_{\geq 1}; \qquad
F(z) = \sum_{n=0}^{\infty} n! z^{-n-1}.
\]

\end{enumerate}

\end{document}
