This isn't homework etc just revision where I do not have the solutions and I have no idea how to approach this question. Any tips or guidance will be greatly appreciated. Thanks.
Let the probability function, $p_n = P(N = n)$, of the frequency r.v. N, belongs to the Panjer’s family of distributions, i.e.
$p_n = (a + \frac{b} {n} ) p_{n-1}$
Prove that the probability generating function of the frequency r.v., namely $P_N(z) = E(z^N)$, satisfies the equation the following differential equation
$P′_N(z) = azP'_N(z) + (a + b)P_N(z)$
Prove that the p.g.f. of the aggregate claims $S = X_1 + X_2 + · · · + X_N$, namely $P_S(z) = E(z^S)$, satisfies the following differential equation
$P′_S(z) = aP_X(z)P'_S(z) + (a + b)P_S(z)P'_X(z)$
Hence show
$P_N(z) = (\frac{1-a} {1-az})^{\frac{b} {a+1}}$
I have tried differentiating this using $ P_N(z) = \sum\limits_{0}^{\infty} p_n z^{n}$ and Panjer's recursion formulae to obtain
$ P_N'(z) = \sum\limits_{0}^{\infty} n p_n z^{n-1} $
$ P_N'(z) = \sum\limits_{0}^{\infty} n (a + \frac{b} {n} ) p_{n-1} z^{n-1} $