As observed in Find the expected value of $E[\frac{1}{(1+X)^2}]$ where X is binomial, there exists a closed-form solution to
$$ E\left[\frac{1}{(1+X)^2}\right] $$
where $X$ is a binomial random variable, i.e $X∼Bin(n,p)$. I greatly thank the author who replied to the previous post (@wolfies)!
Current problem: I am now faced with the problem of finding closed-form solution for
$$ E\left[\frac{1}{(2+X)^2}\right] \text{ where } X\sim Bin(n,p)$$
Any ideas on how to proceed? Is there perhaps a way to extend the closed-form solution proposed by @wolfies to the case where we have $(2+X)$, instead of $(1+X)$ at the denominator?
In the literature, I was able to find some insights inside Chao and Strawderman(1972), i.e. Journal of the American Statistical Association, Vol. 67, No. 338 (Jun., 1972), pp. 429-431.
Nevertheless, an analytical formula is provided for
$$ E\left[\frac{1}{(A+X)^k}\right] \:\text{ where }\: X \sim Bin(n,p) \:\text{ and }\: A \text{ is a constant} $$ only in the setting where $A=1$ and $k=1$, i.e.
$$ E\left[\frac{1}{(1+X)}\right] = \frac{1-q^{(n+1)}}{(n+1)p} $$