# Combinatoric formulas of Generalized Laguerre polynomials

I came up with the following quantity:

$$\sum_{r=0}^p \binom{n}{n-p+r} \binom{2r}{2r-p} L^{(n-p-r)}_{2r}(z) \, ,$$ where $n\ge p$ are integers and $L^{(\alpha)}_m(z)$ is the generalized Laguerre polynomials.

I expect that we can express the last sum in a simple way. I would like to ask if anyone knows of such a result. Thank you in advance.

• Did you look at the classic orthogonal polynomials books? Szego "Orthogonal polynomials" is dated but very good, Mourad "Classic and Quantum orthogonal polynomials" is more modern. – Amir Sagiv Feb 17 '17 at 15:22
• Mathematica does this sum in closed form, but the expression is a DifferenceRoot of a truly awesome messy function involving polynomials that are of high degrees in $n$ and $p$ and themselves involve binomial coefficients and generalized Laguerre polynominals. Not sure this would be progress. – Mark Fischler Feb 18 '17 at 0:58