Meekah
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I have problem with prooving those two identities. Any help would be much appriciated!
Show that:
a)
[tex]\begin{Bmatrix}<br /> <br /> m+n+1\\ m <br /> <br /> \end{Bmatrix}<br /> <br /> = \sum_{k=0}^{m} k \begin{Bmatrix}<br /> <br /> n+k\\k <br /> <br /> \end{Bmatrix}[/tex]
b)[tex] \sum_{k=0}^{n} \begin{pmatrix}<br /> <br /> n\\k <br /> <br /> \end{pmatrix}<br /> <br /> \begin{Bmatrix}<br /> <br /> k\\m <br /> <br /> \end{Bmatrix}<br /> <br /> = \begin{Bmatrix}<br /> <br /> n+1\\m+1 <br /> <br /> \end{Bmatrix} <br /> [/tex]
Where:
\begin{Bmatrix}
k\\m
\end{Bmatrix}
is a Stirling number of the second kind.
Show that:
a)
[tex]\begin{Bmatrix}<br /> <br /> m+n+1\\ m <br /> <br /> \end{Bmatrix}<br /> <br /> = \sum_{k=0}^{m} k \begin{Bmatrix}<br /> <br /> n+k\\k <br /> <br /> \end{Bmatrix}[/tex]
b)[tex] \sum_{k=0}^{n} \begin{pmatrix}<br /> <br /> n\\k <br /> <br /> \end{pmatrix}<br /> <br /> \begin{Bmatrix}<br /> <br /> k\\m <br /> <br /> \end{Bmatrix}<br /> <br /> = \begin{Bmatrix}<br /> <br /> n+1\\m+1 <br /> <br /> \end{Bmatrix} <br /> [/tex]
Where:
\begin{Bmatrix}
k\\m
\end{Bmatrix}
is a Stirling number of the second kind.