Ramanujan's master theorem says that for analytic functions satisfying a certain growth condition, $\displaystyle{\int_0^{\infty}} x^{s-1} \sum_{k=0}^{\infty} \frac{(-x)^k}{k!} \varphi(k) = \Gamma(s) \varphi(-s) \, dx$.
I'm wondering if there is a nice extension to such Mellin transforms but only on part of the positive real line. That is, for $\theta(x)$ denoting the Heaviside step function, $c \in \mathbb{R}^+$
$\displaystyle{\int_0^{\infty}} x^{s-1} \theta(x-c) \sum_{k=0}^{\infty} \frac{(-x)^k}{k!} \varphi(k) \, dx = \displaystyle{\int_c^{\infty}} x^{s-1} \sum_{k=0}^{\infty} \frac{(-x)^k}{k!} \varphi(k) \, dx$.
Using some umbral calculus, where $\text{eval}[a^k] = a_k$,
where $f(x) = \sum_{k=0}^{\infty} a_k \frac{(-x)^k}{k!} = \text{eval}[\sum_{k=0}^{\infty} a^k \frac{(-x)^k}{k!}] =\text{eval}[e^{-ax}] $
$\text{eval}[\displaystyle{\int_c^{\infty}} x^{s-1} e^{-ax} \, dt] = \text{eval}[a^{-s} \Gamma(s,ac)] = \text{eval}[a^{-s}\Gamma(s)] - \text{eval}[a^{-s} \gamma(s,ac)] = a_{-s} \Gamma(s) - c^s \displaystyle{\sum_{k=0}^{\infty}} \frac{a_k (-c)^k}{k!(s + k)}$.
However, I'm wondering if there's maybe a different approach as the sum on the right hand side is not always easy to derive a closed form for, if there is one even.
So the questions would be,
Is there a better way to derive closed form solutions to these "partial" Mellin Transforms?
I couldn't find a lot on tables of Mellin transforms for functions of the type $\theta(x-c) f(x)$, except for $\theta(x-1) f(x)$,$f(x) = x^{\alpha}, x^{\alpha} \ln(x)$, for instance: https://en.wikipedia.org/wiki/Mellin_transform#Table_of_selected_Mellin_transforms.
Can anyone recommend any resources that might have results for functions of this type?
- Is there a way I can determine which kinds of functions it is reasonable to expect a closed form for?