Directly compute the convolution sum of the following two sequences $y(n) = x(n) * h(n)$.
$$
h(n) = \left\{\begin{aligned}
& a^n, & 0 \leqslant n \leqslant N-1 \\
& 0, & \text{others} \\
\end{aligned} \right.
$$
$$
x(n) = \left\{\begin{cases}
\beta^{n-n_0}, & n_0 \leqslant n \\
0, & n \leqslant n_0 \\
\end{cases} \right.
$$
Please express using formulas.

Options:
A. (1) When $n < n_{0}$, $y(n) = 1.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \delta^{n-n_{0}}(n-n_{0}), \quad \delta=\beta
$$
$$
y(n) = \frac{\delta^{n-n_{0}}-\beta^{n+1}}{\delta-\beta}, \quad \delta \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \delta^{n-n_{0}}, \quad \delta=\beta
$$
$$
y(n) = \beta^{n-N-n_{0}} \frac{\delta^{N}-\beta^{N}}{\delta-\beta}, \quad \delta \neq \beta
$$
B. (1) When $n < n_{0}$, $y(n) = 8.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \iota^{n-n_{0}}(n+2), \quad \iota=\delta
$$
$$
y(n) = \frac{\iota^{n+1-n_{0}}-\beta^{n+1}}{\iota-\beta}, \quad \iota \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \iota^{n-n_{0}}, \quad \iota=\delta
$$
$$
y(n) = \beta^{n-N} \frac{\iota^{N}-\beta^{N-1}}{\iota-\beta}, \quad \iota \neq \beta
$$
C. (1) When $n < n_{0}$, $y(n) = 6.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \varepsilon^{n-n_{0}}(n-n_{0}), \quad \varepsilon=\beta
$$
$$
y(n) = \frac{\varepsilon^{n-n_{0}}-\beta^{n+1}}{\varepsilon-\beta}, \quad \varepsilon \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \varepsilon^{n-n_{0}}, \quad \varepsilon=\beta
$$
$$
y(n) = \beta^{n-N-n_{0}} \frac{\varepsilon^{N}-\beta^{N}}{\varepsilon-\beta}, \quad \varepsilon \neq \beta
$$
D. (1) When $n < n_{0}$, $y(n) = 1.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \gamma^{n-n_{0}}(n+2-n_{0}), \quad \gamma=\delta
$$
$$
y(n) = \frac{\gamma^{n+2-n_{0}}-\beta^{n+1-n_{0}}}{\gamma-\beta}, \quad \gamma \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \gamma^{n-n_{0}}, \quad \gamma=\delta
$$
$$
y(n) = \beta^{n+1-N-n_{0}} \frac{\gamma^{N}-\delta^{N}}{\gamma-\delta}, \quad \gamma \neq \delta
$$
E. (1) When $n < n_{0}$, $y(n) = 3.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \beta^{n-n_{0}}(n-1), \quad \gamma=\beta
$$
$$
y(n) = \frac{\gamma^{n-n_{0}}-\beta^{n-n_{0}}}{\gamma-\beta}, \quad \gamma \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \gamma^{n-n_{0}}, \quad \gamma=\beta
$$
$$
y(n) = \beta^{n-N-n_{0}} \frac{\gamma^{N}-\beta^{N}}{\gamma-\beta}, \quad \gamma \neq \beta
$$
F. (1) When $n < n_{0}$, $y(n) = 4.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \zeta^{n-n_{0}}(n-1), \quad \zeta=\beta
$$
$$
y(n) = \frac{\zeta^{n+1-n_{0}}-\beta^{n}}{\zeta-\beta}, \quad \zeta \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \zeta^{n-n_{0}}, \quad \zeta=\beta
$$
$$
y(n) = \beta^{n-N} \frac{\zeta^{N}-\beta^{N-1}}{\zeta-\beta}, \quad \zeta \neq \beta
$$
G. (1) When $n < n_{0}$, $y(n) = 7.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \theta^{n-n_{0}}(n+1), \quad \theta=\beta
$$
$$
y(n) = \frac{\theta^{n+1-n_{0}}-\beta^n}{\theta-\beta}, \quad \theta \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \theta^{n-n_{0}}, \quad \theta=\beta
$$
$$
y(n) = \beta^{n-N-n_{0}} \frac{\theta^{N}-\beta^{N}}{\theta-\beta}, \quad \theta \neq \beta
$$
H. (1) When $n < n_{0}$, $y(n) = 0.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \alpha^{n-n_{0}}(n+1-n_{0}), \quad \alpha=\beta
$$
$$
y(n) = \frac{\alpha^{n+1-n_{0}}-\beta^{n+1-n_{0}}}{\alpha-\beta}, \quad \alpha \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \alpha^{n-n_{0}}, \quad \alpha=\beta
$$
$$
y(n) = \beta^{n+1-N-n_{0}} \frac{\alpha^{N}-\beta^{N}}{\alpha-\beta}, \quad \alpha \neq \beta
$$
I. (1) When $n < n_{0}$, $y(n) = 5.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \eta^{n-n_{0}}(n+1-n_{0}), \quad \eta=\gamma
$$
$$
y(n) = \frac{\eta^{n+1-n_{0}}-\gamma^{n+1-n_{0}}}{\eta+\beta}, \quad \eta \neq \gamma
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \eta^{n-n_{0}}, \quad \eta=\gamma
$$
$$
y(n) = \beta^{n+1-N-n_{0}} \frac{\eta^{N}-\gamma^{N}}{\eta+\gamma}, \quad \eta \neq \gamma
$$
J. (1) When $n < n_{0}$, $y(n) = 2.$
(2) When $n_{0} \leq n \leq n_{0} + N - 1$,
$$
y(n) = \alpha^{n-n_{0}}(n+1-n_{0}), \quad \alpha=\gamma
$$
$$
y(n) = \frac{\alpha^{n+1-n_{0}}-\beta^{n+1-n_{0}}}{\alpha+\beta}, \quad \alpha \neq \beta
$$
(3) When $n > n_{0} + N - 1$,
$$
y(n) = N \alpha^{n-n_{0}}, \quad \alpha=\gamma
$$
$$
y(n) = \beta^{n+1-N-n_{0}} \frac{\alpha^{N}-\beta^{N}}{\alpha+\beta}, \quad \alpha \neq \beta
$$
Answer: H