# Combinatorial problem: triple binomial product related to squared Laguerre polynomials

Context

Hydrogenic wavefunctions [1] include a factor given by Laguerre polynomials [2]. These wavefunctions are often encountered in a first course in quantum mechanics. They also appear in subsequent courses in atomic physics. These wavefunctions are also applicable to the study of two-electron atoms such as helium. In the course of my studies, I realized that I can not determine how to derive the identity below. I have checked that it is true using software, but I can not show it.

Question

Given $$n\in\mathbb{N}$$ and $$\ell\in\left\{\mathbb{N}\mid n>\ell\geq 0\right\}$$, how can the following identity be proven: \begin{align*} \frac{2 \,n \left(n + \ell\right)!} {(n - \ell - 1)!} &= \sum_{j=0}^{2\left[n - \ell - 1\right]} \frac{(-1)^{j} \left(2\,\ell+2+j\right)! }{j!} \sum_{k=0}^{n - \ell - 1 - \left|n - \ell - 1-j \right| } \\ &\qquad\qquad \times {n + \ell \choose k - \left[ j-n + \ell + 1 \right]\left[1-u(j - n + \ell + 1 ) \right]} \\ & \qquad\qquad\times {n + \ell \choose j - k - \left[j - n + \ell + 1\right]\left[1 +u(j - n + \ell + 1)\right] } \\ & \qquad\qquad\times {j \choose k + ( j - n + \ell + 1 )\,u(j - n + \ell + 1) } \end{align*} Above, $$u$$ is the Heaviside-step function.

My attempt

I looked in CRC and I looked online trying to find a form similar to what I have here. On [4] I found Dixon's identity, which does include a product of three binomial coefficients. However, Dixon's identity does not meet my needs. This is so since in my problem there is a factor $$(-1)^j$$, which does not appear in Dixon's identity. I have searched this site and found [5]. I found at least one identity that includes the product of two binomials (e.g., cf., [3]); however, I could not find any identities with the product of three binomials. I have looked on this site for products of three binomials. In [5], I find a question with three binomial coefficients whose problem statement includes several leads, which will be explored in due time. In [6], I also have found a question with a similar expression to mine and the question in [6] also includes a factor $$(-1)^j$$.

Bibliography

• I know this is an old post, but just incase - I recently came across a similar form in my own problem (see here. I have not checked the indices, but perhaps its related. Commented Aug 12, 2022 at 16:12