$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f \rightarrow g)^c}]] $$
Now i want this summation to be grouped in terms of $ln(g(x))$ such that:
$$ D = \sum_k \sum_j \ln(g(x))^{k-c-j}F(k,j) + \sum_j H(j) $$
Or something along those lines
My work:
$$ D = A + B $$
$$ A = f'(x)\sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j+1}(g(x))f_c^{(j)} X_{n,k(f\rightarrow g)^c}] $$
$$ B = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f \rightarrow g)^c}]] $$
$$ A = f'(x)\sum_{k=c+1}^{n}\sum_{j=0}^{k-c-1}[{k-c-1 \choose j}\ln^{k-c-j}(g(x)) f'(x) f_c^{(j)} X_{n,k-1(f\rightarrow g)^c}] $$
But i have a strong feeling i messed up the summation here in this part
