A short proof of Mathar's 2013 recurrence conjecture for the Laguerre sequence~A025166

Abstract

For the OEIS sequence A025166, defined by $a(n)=-n!2^n{L}_n(1/2)$ where $L_n$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence \[ a(n) + (-4n+3)\, a(n-1) + 4(n-1)^{2}\, a(n-2) \;=\; 0, \qquad n \ge 2. \] We give a one-page proof. The exponential generating function $F(x)=-\exp (-x/(1-2x))/(1-2x)$ satisfies the first-order linear ODE $(1-2x{)}^2{F}^{\prime }(x)=(1-4x)F(x)$, and Mathar's recurrence then falls out by reading off the coefficient of $x^n/n!$. Both steps are short. The supplementary archive includes a SymPy script which checks the ODE identically and the recurrence numerically up to $n=5000$.

arXiv comment8 pages