Combinatorics and Graph Theory · Problem 1.8.10

Fundamentals — David Guichard

Mathematics — Combinatorics · stirling-numbers-extensions

Problem

We have defined [nk]\left[\begin{smallmatrix}n\\k\end{smallmatrix}\right] and {nk}\left\{\begin{smallmatrix}n\\k\end{smallmatrix}\right\} for n,k≥0n, k \geq 0. We want to extend the definitions to all integers. Without some extra stipulations, there are many ways to do this. Let us suppose that for n≠0n \neq 0 we want $\left[\begin{smallmatrix}n 0\end{smallmatrix}\right] = \left[\begin{smallmatrix}0\ \end{smallmatrix}\right] = \left\{\begin{smallmatrix}n 0\end{smallmatrix}\right\} = \left\{\begin{smallmatrix}0\ \end{smallmatrix}\right\} = 0,andwewanttherecurrencerelationsofequation1.8.1andintheorem1.8.3tobetrue.Showthatundertheseconditionsthereisauniquewaytoextendthedefinitionstoallintegers,andthatwhenthisisdone,, and we want the recurrence relations of equation 1.8.1 and in theorem 1.8.3 to be true. Show that under these conditions there is a unique way to extend the definitions to all integers, and that when this is done, \left\{\begin{smallmatrix}n k\end{smallmatrix}\right\} = \left[\begin{smallmatrix}-k -n\end{smallmatrix}\right]forallintegers for all integers nand and k.Thus,theextendedtableofvaluesforeither. Thus, the extended table of values for either \left[\begin{smallmatrix}n k\end{smallmatrix}\right]or or \left\{\begin{smallmatrix}n k\end{smallmatrix}\right\}willcontainallthevaluesofboth will contain all the values of both \left[\begin{smallmatrix}n k\end{smallmatrix}\right]and and \left\{\begin{smallmatrix}n k\end{smallmatrix}\right\}$.

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