In the book Concrete Mathematics, chapter 2, section 2.2 — sums and recurrences, page 26 (2nd edition), the authors talk about the following example:
Given the general recurrence
$$ R(0) = \alpha $$
$$ R(n) = R(n-1) + \beta + \epsilon n $$
The authors generalize the recurrence relation to:
$$ R(n) = A(n)\alpha + B(n)\beta + C(n)\epsilon $$
Employing the Repertoire Method, the authors plug in simple functions of $n$ in order to determine $A(n), B(n), C(n)$. So they discover:
Setting $R(n) = 1$ implies $\alpha = 1, \beta = 0, \epsilon = 0 \implies A(n) = 1$.
Setting $R(n) = n$ implies $\alpha = 0, \beta = 1, \epsilon = 0 \implies B(n) = n$.
Setting $R(n) = n^2$ implies $\alpha = 0, \beta = -1, \epsilon = 0 \implies C(n) = \frac{n^2 + n}{2}$.
Values for the first couple of terms of the recurrence:
$$
\begin{eqnarray*}
R(0) &=& \alpha
\\ R(1) &=& \alpha + \beta + \epsilon
\\ R(2) &=& \alpha + 2\beta + 3\epsilon
\\ R(3) &=& \alpha + 3\beta + 6\epsilon
\\ R(4) &=& \alpha + 4\beta + 10\epsilon
\\ R(5) &=& \alpha + 5\beta + 15\epsilon
\end{eqnarray*}
$$
I do not understand what is the process through which the values for $\alpha$, $\beta$, and $\epsilon$ are implied. I would like some help with that. Where exactly do we look and what do we math them against to see what they have to be?
Best Answer
Put $R_n=1$ (for all $n$; hence also $R_0$ and $R_{n-1}$ should be set equal to 1) in (2.7): $$ 1 = \alpha, \quad 1 = 1 + \beta + \gamma n. $$ The first equation tells us $\alpha$ right away, and the second equality holds for all $n$ iff $\beta=\gamma=0$.
Then put $R_n=n$ (hence $R_0=0$ and $R_{n-1}=n-1$) in (2.7): $$ 0 = \alpha, \quad n = (n-1)+\beta + \gamma n. $$ Here $\beta=1$ and $\gamma=0$ is required for the identity to hold for all $n$ (compare coefficients for the constant terms and for the $n$-terms separately).
Etc.