Let $$g(x) = \log f(x)$$ where $$f(x)$$ is a twice differentiable positive function on $$(0, \infty)$$ such that $$f(x + 1) = x f(x)$$. Then, for $$N = 1, 2, 3,...,$$
$$g''\left(N + \frac{1}{2}\right) - g''\left(\frac{1}{2}\right) =$$

A

$$-4\left\{1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{(2N - 1)^2}\right\}$$

B

$$4\left\{1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{(2N - 1)^2}\right\}$$

C

$$-4\left\{1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{(2N + 1)^2}\right\}$$

D

$$4\left\{1 + \frac{1}{9} + \frac{1}{25} + ... + \frac{1}{(2N + 1)^2}\right\}$$