8.2 Given the sequence h_{0}, h_{1}, h_{2}, ..., define its difference sequence by

Δh_{0}, Δh_{1}, Δh_{2}, ... where Δh_{n} = h_{n+1} − h_{n}

and its pth order difference sequence by Δ^{p}h_{0}, Δ^{p}h_{1}, Δ^{p}h_{2}, ... where Δ^{p}h_{n} = Δ(Δ^{p−1}h_{n})

Ex: h_{n} = n^{2} − n + 1

Thm 8.2.1 Suppose h_{n} = a_{p}n^{p} + a_{p−1}n^{p−1} + ... + a_{1}n + a_{0}. Then Δ^{p+1}h_{n} = 0 ∀n Proof by induction on p: p = 0:

Note that the set of all complex-valued sequences form a vector space and Δ is a linear transformation on this vector space.

Defn: The 0th diagonal of the difference table for h_{n} is h_{0} = Δ^{0}h_{0}, Δ^{1}h_{0}, Δ^{2}h_{0}, ...

Note the 0th diagonal of the difference table for h_{n} determines h_{n}

Lemma: Suppose the 0th diagonal of the difference table for h_{n} is 0, 0, ..., 0, 1, 0, 0, ... where 1 is preceded by exactly p

zeros. Then h_{n} = ( \begin{matrix}n \\ p \end{matrix} )

Thm 8.2.2: Suppose 0th diagonal of the difference table for h_{n} is c_{0}, c_{1}, ..., c_{p}, 0, 0, 0, ....

Then h_{n} = c_{0} ( \begin{matrix}n \\ 0 \end{matrix} ) + c_{1} ( \begin{matrix}n \\ 1 \end{matrix} ) + ... + c_{p} ( \begin{matrix}n \\ p \end{matrix} ).

Thm 8.2.3: Suppose 0th diagonal of the difference table for h_{n} is c_{0}, c_{1}, ..., c_{p}, 0, 0, 0, ....

Then Σ_{k=0}^{n}h_{k} = c_{0} ( \begin{matrix}n + 1 \\ 1 \end{matrix} )+c_{1} ( \begin{matrix}n + 1 \\ 2 \end{matrix} )+...+c_{p} ( \begin{matrix}n + 1 \\ p + 1 \end{matrix} ).

Ex: Find Σ_{k=0}^{n}k^{3} = Let h_{n} = n^{p}

Then the 0th diagonal of the difference table for h_{n} is c(p, 0), c(p, 1), ..., c(p, p), 0, 0, 0, ....

c_{0} ( \begin{matrix}n + 1 \\ 1 \end{matrix} ) + c_{1} ( \begin{matrix}n + 1 \\ 2 \end{matrix} ) + ... + c_{p} ( \begin{matrix}n + 1 \\ p + 1 \end{matrix} ).