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\title{General Mathematics and Computational Science I}
\author{Exercise 11}
\date{October 18, 2005}
\maketitle

\begin{enumerate}

\item Recall that the generalized binomial coefficients are defined as
the coefficients in the formal power series
\begin{equation*}
  (1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} \, x^k \,.
\end{equation*}

Show that
\begin{equation*}
  \binom{\alpha}{k+1}
  = \frac{\alpha-k}{k+1} \, \binom{\alpha}{k}
\end{equation*}
for every real number $\alpha$, and $0 \leq k$.  

\item Use the method of generating functions to find a closed form
expression for the members of the sequence
\begin{gather*}
  c_0 = 1 \,, \\
  c_{n+1} = \sum_{k=0}^n c_k \, c_{n-k} \,.
\end{gather*}

\emph{Hint:} Your answer will involve generalized binomial
coefficients with $\alpha=\tfrac12$, see Question~1 above.  You may
leave your answer in terms of these coefficients; there is no need to
further expand the expressions although you may want to check the
first couple of terms to see whether your answer is correct.

\end{enumerate}
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