Concrete Mathematics: Sums and Recurrences

Question

In the book Concrete Mathematics(Graham, Knuth and Patashnik) section 2.2 Sums and Recurrences, is given:

\begin{align} S_0=a_0 \\S_n = S_{n-1}+a_n, \end{align}

General Form(2.7):

\begin{align} R_0 = \alpha \\R_n = R_{n-1} + \beta + \gamma n, \end{align}

In general the solution can be written in the form (2.8):

\begin{align}R_n = A(n) \alpha + B(n) \beta + C(n) \gamma \end{align}

Here is where I get confused:

Setting $\\R_n = 1$ implies that $\alpha$ = 1, $\beta$ = $\mathbb{0}$ and $\gamma$=$\mathbb{0}$

Setting $\\R_n = n$ implies that $\alpha$ = $\mathbb{0}$, $\beta$ = $\mathbb{1}$ and $\gamma$=$\mathbb{0}$

Setting $\\R_n = n^2$ implies that $\alpha$ = $\mathbb{0}$, $\beta$ = $\mathbb{-1}$ and $\gamma$=$\mathbb{2}$

Why setting this values for $\\R_n$ implies that values for $\alpha$, $\beta$ and $\gamma$

Thanks in advance,

Daniel.

Answer 1

The recurrence for $R_n$ is a particular case of the recurrence for $S_n$ when $a_n=\beta+\gamma\,n$. In general, $\{a_n\}$ could be any sequence, not necessarily of the form $\beta+\gamma\,n$. Fnding $\beta$ and $\gamma$ does not make sense; consider for example $S_n=S_{n-1}+n^2$. However, $\alpha=a_0$.