I am working on simplifying a complex expression that arises from a quantum mechanics problem involving matrix elements. The expression is given by an integral involving Laguerre polynomials, which I have attempted to simplify using a known formula. The integral I started with is:
$$ \langle n,0| r^2 |n',2\rangle = C \int_{0}^{\infty} t^6 e^{-t\left(\frac{1}{n} + \frac{1}{n'}\right)} L_{n-1}^{(1)}\left(\frac{2t}{n}\right) L_{n'-3}^{(5)}\left(\frac{2t}{n'}\right) dt $$
Using the integral formula from this link:
http://functions.wolfram.com/05.08.21.0009.01
I have identified the constants as follows:
$\alpha = 7$
$a = \frac{2}{n}$
$b = \frac{2}{n'}$
$p = \frac{1}{n'} + \frac{1}{n}$
$m = n - 1$
$n = n' - 3$
$\lambda = 1$
$\beta = 5$
After substituting these constants into the formula, I have simplified the prefactor before the sums:
$$ \Gamma(7) \cdot \frac{\text{Pochhammer}[2, n-1]}{(n-1)!} \cdot \frac{\text{Pochhammer}[6, n'-3]}{(n'-3)!} $$
as:
$$ 6! \cdot n \cdot \frac{n'(n'^2 - 1)(n'^2 - 4)(n' - 3)}{5!} $$
and then to the multiplicative coefficient:
$$ 6 \cdot \left(\frac{nn'}{n+n'}\right)^7 \cdot n \cdot n' \cdot (n'^2 - 1) \cdot (n'^2 - 4) \cdot (n' - 3) $$
I am now looking for ways to simplify the sums that follow this coefficient (if there are any ways to do so). The sums involve Pochhammer symbols and powers of the variables $a/p$ and $b/p$, and I am unsure how to proceed with their simplification. Here is the part with the sums:
$$ \sum_{j=0}^{n-1} \left[\frac{(\text{Pochhammer}[1-n, j] \text{Pochhammer}[7, j])}{(\text{Pochhammer}[2, j] j!)} \left(\frac{\frac{2}{n}}{\frac{1}{n'} + \frac{1}{n}}\right)^j \sum_{k=0}^{n'-3} \left[\frac{(\text{Pochhammer}[3-n', k] \text{Pochhammer}[7 + j, k])}{(\text{Pochhammer}[6, k] k!)} \left(\frac{\frac{2}{n'}}{\frac{1}{n'} + \frac{1}{n}}\right)^k\right]\right] $$
I would greatly appreciate any insights or suggestions on how to approach the simplification of these sums or any general advice on handling such expressions.