I am trying to calculate the simple Laguerre polynomial for the half-integer orders.
From https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch7/laguerre.html, I can get the Laguerre polynomials for integer orders, but I am trying to find a simpler formula for the half-integer orders. For $s=1,3,5,\ldots$, I know that $$\mathbb{L}_{\frac s2} (x) = {}_1\mathbb{F}_{1}(-\frac{s}2, 1, x)$$ but is there a simpler way to get to the formula for small $s$? I really only need the first few $s$. Thanks!
