Hello I was wondering what was the function $f$ defines like this :

Let $f(x)$ a continuous and differentiable function such that : $$f(x)=\sum_{k=0}^{\infty}\frac{f'(k)}{k!}(-x)^k$$

In fact can't solve it but it makes a connection between Ramanujan's Master theorem and Frullani's integral via the Fundamental theorem of calculus I explain :

We have :

$$\int_{0}^{\infty}x^{-s-1}f(x)dx=\Gamma{(-s)}f'(s)$$

Or : $$\int_{0}^{\infty}\frac{x^{-s-1}f(x)}{\Gamma{(-s)}}dx=f'(s)$$

Now we use the Fundamental theorem of calculus to get : $$\int_{0}^{s}\int_{0}^{\infty}\frac{x^{-s-1}f(x)}{\Gamma{(-s)}}dxds=f(s)-f(0)$$

Now we take the limit to get : $$\lim_{s\to\infty}\int_{0}^{s}\int_{0}^{\infty}\frac{x^{-s-1}f(x)}{\Gamma{(-s)}}dxds=f(\infty)-f(0)$$

$$\int_{0}^{\infty}\frac{f(ax)-f(bx)}{ln(\frac{a}{b})x}$$

So my question is what is the function $f(x)$ , there exists a closed form to this ,is it trivial or not ?

Thanks

Ps:I know it's not very rigorous but I think it's interesting