Let $f(z) = \sum\_{j=0}^{\infty}a\_jz^j$ be the Maclaurin expansion of a function $f(z)$ analytic at the origin. Prove each of the following statements.

$(a)$ $\sum\_{j=0}^{\infty}a\_jz^{2j}$ is the Maclaurin expansion of $g(z) : = f(z^2)$

$(b)$ $\sum\_{j=0}^{\infty}a\_jz^{m+j}$ is the Maclaurin expansion of $H(z) : = z^mf(z)$

Hint: You may use the uniqueness of Taylor series( the following theorem) when solving the questions above.

Theorem: If $\sum\_{j=0}^{\infty}a\_j(z-z\_0)^{j}$ converges to $f(z)$ in some circular neighborhood of $z\_0$ (that is, the radius of its circle of convergence is nonzero), then

$$a\_j = \dfrac{f^{(j)}(z\_0)}{j!} $$
where $j = 0,1,2,....$

Consequently $\sum\_{j=0}^{\infty}a\_j(z-z\_0)^{j}$ is the Taylor expansion of $f(z)$ around $z\_0$

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I don't understand how should I use the Theorem to solve the problem, any help would be appreciated