%--------------------------------------------------------------
\begin{frame}
\begin{center}
\Huge Identities for Roots of Macdonald function
\end{center}
\end{frame}

\begin{frame}
\scriptsize
\frametitle{Sketch of Main Results}
\begin{itemize}
\item
Substitute $\color{eqncolor}\Psi_\ell(t,r) = 
\sum_{k=0}^\ell \frac{1}{r^k} c_{\ell k} f^{(\ell-k)}(t-r)$ into
$$\color{eqncolor}
\partial_t \Psi_\ell(t,r) + \partial_r \Psi_\ell(t,r)= 
\sum_{n=1}^\ell \frac{b_{\ell n}}{r^2}
\int_0^t e^{\frac{b_{\ell n}}{r}(t-t')} \Psi_\ell(t',r)dt'
$$
\item Result:
$$\color{eqncolor}
-\sum_{k=1}^\ell \frac{k}{r^{k+1}} c_{\ell k} f^{(\ell-k)}(t-r)
= \sum_{n=1}^\ell \frac{b_{\ell n}}{r^2} 
\sum_{k=0}^\ell
\frac{1}{r^k} c_{\ell k} 
I^{(\ell-k)}[b_{\ell n},r,f]
$$

\item 
Here $\color{eqncolor}I^{(p)}[b,r,f] \equiv \int_0^t 
e^{\frac{b}{r}(t-t')} f^{(p)}(t'-r) dt'$

\item
Work into form (argument sketched later...integration by parts)
$$\color{eqncolor}
0 = \sum_{k=1}^\ell r^{-(k+1)} E_k f^{(\ell-k)}(u),\quad
E_k = k c_{\ell k} + \sum_{n=1}^\ell b_{\ell n} \sum_{q=1}^k c_{\ell,q-1}
(b_{\ell n})^{k-q}
$$
\end{itemize}
\end{frame}
\begin{frame}
\scriptsize
\frametitle{Sketch of Main Results}
\begin{itemize}
\item
Starting with (last line of last slide)
$$\color{eqncolor}
0 = \sum_{k=1}^\ell r^{-(k+1)} E_k f^{(\ell-k)}(u),\quad
E_k = k c_{\ell k} + \sum_{n=1}^\ell b_{\ell n} \sum_{q=1}^k c_{\ell,q-1}
(b_{\ell n})^{k-q},
$$
isolate terms $\color{eqncolor}E_k f^{(\ell-k)}(u)$ with operator $\color{eqncolor}Q
= (\partial_t + \partial_r)r^2$.
\item Example ($\color{eqncolor}\ell = 3$):
\begin{align*}\color{eqncolor}
Q\big[
  \frac{1}{r^2} E_1 f''(u) +
  \frac{1}{r^3} E_2 f'(u) +
  \frac{1}{r^4} E_3 f(u) \big] \color{eqncolor} & = 
\color{eqncolor}-\frac{1}{r^2}E_2 f'(u)-\frac{2}{r^3}E_3 f(u)\\
  \color{eqncolor} Q^2\big[
  \frac{1}{r^2} E_1 f''(u) +
  \frac{1}{r^3} E_2 f'(u) +
  \frac{1}{r^4} E_3 f(u) \big] \color{eqncolor}& \color{eqncolor}= \frac{6}{r^2}E_3 f(u) 
\end{align*}
\item
Profile $\color{eqncolor}f$ arbitrary 
$\color{eqncolor}\implies E_3 = 0 \implies E_2 = 0 \implies E_1 = 0$.
\end{itemize}
\end{frame}
