Functions and things and integrals

Define the functions $\{ f_n\} _ {n \geq 0}$ as follows:

(1) $f_0(x) = \Gamma \left(\frac{x+1}{x}\right) \text{, and}$

(2) $f_j(x)=x^2 \frac{d f_{j-1}(x)}{d x} \text{, for } j>0.$

$\underset{x\to \infty }{\text{lim}}f_6(x)=\frac{\gamma ^4 \pi ^2 a}{b}+\frac{\gamma ^2 \pi ^4 c}{d}+\frac{\pi ^6 g}{h}+j \gamma \left(2 \gamma ^2+\pi ^2\right) \zeta (3)+k \zeta (3)^2+m \gamma \zeta (5)+\gamma ^6$

where $a,b,c,d,g,h,j,k,m$ are positive integers. Submit $a + b + c + d + g + h + j + k + m$ .

Bonus: Find closed form for

$\underset{x\to \infty }{\text{lim}}f_k(x), k > 0$

in terms of $k, \pi, \gamma$ (the Euler–Mascheroni constant), and the zeta function $\zeta$ with positive odd integer arguments.