Eigenvectors of a tridiagonal stochastic matrix I'm looking for the eigenvectors of this matrix:
\begin{equation}
\nonumber
M =
\frac{1}{N} \left( \begin{array}{ccccccccc}
 0 & 1 &&&&&&&\\ 
N &  0 & 2&&&&&&& \\
&N-1 &  0 & 3&&&&&& \\
&&N-2 &  0& \ldots&&&&& \\
&&&\vdots & \ddots &&&&& \\
&&&&& 0 & N-2 && \\
&&&&&3 & 0 &N-1 & \\
&&&&&&2 & 0&N  \\
&&&&&&& 1 &0 \\
\end{array} \right) 
\end{equation} 
where only nonzero entries are shown. 
This matrix has size $(N+1) \times (N+1)$. Its superdiagonal entries grow linearly from $1$ to $N$, and the subdiagonal entries decrease linearly from $N$ to $1$. 
 A: Let $T$ be a tridiagonal matrix of order $m\times m$ such that the entries
$T_{i+1,i}$ and $T_{i,i+1}$ are positive for all $i$. Let $p_r$ be the characteristic
polynomial of the leading principal $r\times r$ submatrix of $T$ and set $p_0=1$.
Then the polynomials $p_1,\ldots,p_m$ satisfy a three-term recurrence, and it follows
that if $\theta$ is an eigenvalue of $T$, then the vector with entries
\[
  p_0(\theta),\ldots,p_{n-1}(\theta)
\]
is an eigenvector for $T$ with eigenvalue $\theta$. (All this works for any tridiagonal
matrix $T$ such that $T_{i+1,i}T_{i,i+1}>0$ for all $i$. This is part of the theory of
orthogonal polynomials.)
For your choice of $T$, the eigenvalues are the integers $n-2k$ for $k=0,\ldots,n$
and the polynomials $p_r$ are known as Krawtchouk polynomials. We have (following wikipedia
for example) that
\[
  p_r(t) = \sum_{i=0}^r -1)^i \binom{t}{i} \binom{n-t}{r-i}.
\]
The best reference I know for Krawtchouk polynomials is MacWilliams and Sloane's book on coding theory, I could not find anything really useful online.
