Functions in the Ramanujan master Theorem. Ramanujan Master theorem states that -
Suppose you have a function 

$$F(x)=\sum_{k=0}^\infty \frac {w(k)(-x)^k}{k!}$$ 

for some for some function (say analytic or integrable) $w(x)$.
Then  

$$I=\int_0^\infty x^{n-1}F(x)dx= \Gamma(n)w(-n).$$

Here I am trying to find out some interesting functions $w(x)$ such that $I$ has interesting results . For instance $w(x)=1$ gives $I=e^x$. Like wise can please anyone give me some interesting cases $w(x)$ and $I$.
 A: As an important application, this formula is used to prove that the values of the Riemann zeta function at negative integers are $\zeta(1-n)=-B_n/n$ where $B_n$ are the Bernoulli numbers. In that case, one takes $F(x) = x/(e^x-1)$, the usual generating function for the Bernoulli numbers.
The formula can be phrased by stating vaguely that

The values at integers of the Mellin transform of a function are its Taylor coefficients.

It is not necessary that the Taylor coefficients come from an analytic function $\omega$, they can be any sequence $a_n$, so long as the resulting function $F$ is nice enough. In this case, it is a consequence of the formula that there is a canonical choice of a function interpolating the function $1-n \mapsto a_n$, namely the Mellin transform of $F$, divided by $\Gamma$. If you really think about this, it is quite a surprising thing!
A: The  first simple example is a well known relationship between gamma function and beta function
Let $m,n>0$ and set $x=\frac{y}{1+y}$ to obtain
$$\int_{0}^{1} x^{m-1} (1-x)^{n-1}dx=\int_{0}^{\infty} y^{m-1} (1+y)^{-m-n}dy$$
From the binomial series,
$$(1+y)^{-r}=\sum_{k=0}^{\infty}\frac{\Gamma(k+r)}{\Gamma(r)k!}(-y)^k,  |y|<1$$
we find that
$$\color{blue}{w(s)=\frac{\Gamma(s+m+n)}{\Gamma(m+n)}}$$
using Ramanujan's master  theorem (R.M.T ) we get the following well known representation of beta function $B(m,n)$.
$$B(m,n)=\int_{0}^{1} x^{m-1} (1-x)^{n-1}dx=\Gamma(m)w(-m)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$
You can also find the Mellin transformation of a Bessel function $ J_\alpha(\sqrt{x}).$ using R.M.T
$$I=\int_{0}^{\infty} x^{\beta-1} \bigg(\frac{\sqrt{x}}{2}\bigg)^{\alpha} \frac{1}{\Gamma(1+\alpha)} \sum_{n=0}^{\infty} \frac{1}{(1+\alpha)_n}\frac{(\frac{-x}{4})^n}{n!} dx$$
$I$ can be expressed as 
$$I=\frac{dx}{x}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}w(n)x^{n+\beta+\frac{\alpha}{2}}$$
where 
$$\color{blue}{w(n)=\frac{1}{2^{\alpha+2n}\Gamma(n+\alpha+1)}}$$
then
$$\int_{0}^{\infty}J_\alpha(\sqrt{x}) x^{\beta-1} dx =2^{2\beta}\frac{\Gamma(\beta+\frac{\alpha}{2})}{\Gamma(1+\frac{\alpha}{2}-\beta)}.$$
Another example is the following integral
$$I=\int_{0}^{\infty}x^{p-1}\bigg(\frac{2}{1+\sqrt{1+4x}}\bigg)^n dx$$
for this integral
$$\color{blue}{w(p)=\frac{n\Gamma(n+2p)}{\Gamma(n+p+1)}}$$
then
$$I=\frac{n\Gamma(p)\Gamma(n-2p)}{\Gamma(n-p+1)}$$
An important application of R.M.T is in quantum field theory,
For instance  R.M.T is used to  evaluate Momentum Space Integrals
$$G=\int \frac{d^Dq}{i \pi^{D/2}}\frac{1}{[q^2]^{a_1}[(p-q)^2]^{a_2}}$$
where 
$$\color{blue}{w(k)=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}(p^2)^n \bigg(\frac{D}{2}+n \bigg)_k y^{k+n+a_2}}$$
A: Expand $(1+x)^{-m}$ in its Maclaurin series $\sum_{k=0}^{\infty}\frac{\Gamma(m+k)}{\Gamma(m)}\frac{(-x)^{k}}{k!}=\frac{1}{\Gamma(m)}\sum_{k=0}^{\infty}\Gamma(m+k)\frac{(-x)^{k}}{k!},$ where  $w(k)=\Gamma(m+k).$
Expanding $\frac{1}{1+x}$ into a geometric series $\sum_{k=0}^{\infty}(-x)^{k}.$  Here $w(k)=k!.$ 
