Help with computing the $n$-th derivative of $f(x)=e^{-cx}\sum_{k=1}^{\infty}\frac{c^k}{(k+m)(k-1)!}(x-x_u)^k$

Solution 1:

I inserted your series into WolframAlpha and it gave me this output (I need to check if it is true): $$ f(x)=e^{-cx}\sum_{k=1}^{\infty}\frac{c^k(x-x_u)^k}{(k+m)(k-1)!} = \frac{(-1)^{m+1}e^{-cx}}{c^m(x-x_u)^m}\gamma(m+1, -c(x-x_u)), $$ where $\gamma(a, x)$ is the lower incomplete gamma function.

For computing the $n$-th derivative of a product of functions, you can use the generalized Leibniz rule, where $$ (f_1f_2\dots f_m)^{(n)} = \sum_{k_1+k_2+\dots +k_m=n}\binom{n}{k_1, k_2, \dots, k_m}\prod_{1 \leq t \leq m}f_t^{(k_t)} $$ being $\sum_{t=1}^m k_t = n$ and $$ \binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1!k_2!\dots k_m!}. $$

It will be useful to consider: $$ \begin{aligned} &\frac{\mathrm{d}^k\phantom{x}}{\mathrm{d}x^k}\left(e^{-cx}\right) = (-c)^ke^{-cx}, \\ &\frac{\mathrm{d}^k\phantom{x}}{\mathrm{d}x^k}\left(\frac{1}{(x-x_u)^m}\right) = \frac{(m+k-1)!}{(m-1)!}\frac{(-1)^k}{(x-x_u)^{m+k}}. \end{aligned} $$

For the incomplete gamma function, as $m+1$ is an integer, we can compute it (following this source) as $$ \gamma(m+1, -c(x-x_u)) = m!\left[1-e^{c(x-x_u)}\sum_{l=0}^m\frac{(-1)^lc^l(x-x_u)^l}{l!}\right]. $$

Its $k$-th derivative can be computed using again the generalized Leibniz rule, where $$ \begin{aligned} &\frac{\mathrm{d}^k\phantom{x}}{\mathrm{d}x^k}\left(e^{c(x-x_u)}\right) = c^ke^{c(x-x_u)}, \\ &\frac{\mathrm{d}^k\phantom{x}}{\mathrm{d}x^k} \left(\sum_{l=0}^m\frac{(-1)^lc^l(x-x_u)^l}{l!}\right) = \sum_{l=k}^m\frac{(-1)^lc^ll!(x-x_u)^{l-k}}{(l-k)!}. \end{aligned} $$

If you want, I can go further with the expressions. Hope it is clear (and do not have any typos)!

Solution 2:

Consider this answer, and write your function not as you did but as

$$\partial^{n}f(x) = \sum_{k = 1}^{+\infty} \frac{1}{(k+m)(k-1)!}\frac{\text{d}^n}{\text{d}x^n}\color{red}{\left[e^{-cx}(c(x - x_u))^k\right]}$$

The red term is nothing but $$(-1)^k\color{blue}{e^{-cx}(c(x_u-x))^k)}$$

Where the blue term is the one analysed in the liked question above. Follow the same procedure, and then you will get the $nth$ derivative expression.

Further Details

Copying from the other answer: $$\begin{align} & \text{when}\quad n\le k: \frac{d^nf_k(x)}{dx^n} = (-1)^n\sum_{i=0}^{n} \binom n i (c)^{n-i}[k]_if_{k-i}(x) \\ \end{align}$$

This ensures you the closed expression for that blue term, whence the derivation is ended. You can then write down glueing all the pieces:

$$\partial^nf(x) = \sum_{k=1}^{+\infty}\frac{1}{(k+m)(k-1)!}(-1)^{k+n}\sum_{i=0}^{n} \binom n i (c)^{n-i}[k]_if_{k-i}(x)$$

Where we define $[k]_m = k(k-1)\cdots(k-m)$ for convenience in notations, also $[k]_0 = 1$ and $[k]_1 = k$).