A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615

Tong Niu

paper

A short proof of Mathar's 2013 recurrence conjecture for the Meixner sequence A214615

arXiv:2605.03170

Abstract

For the OEIS sequence A214615, defined by $a (n) = M_{n} (1)$ where $M_{n}$ is the $n$ -th Meixner polynomial satisfying $M_{n + 1} (x) = x M_{n} (x) - n^{2} M_{n - 1} (x)$ , R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2 P-recursive recurrence $a (n) - a (n - 1) + (n - 1)^{2} a (n - 2) = 0$ for $n \geq 2$ . We give a one-page proof. The exponential generating function $F (t) = exp (arctan t) / 1 + t^{2}$ satisfies the first-order linear ODE $(1 + t^{2}) F^{'} (t) = (1 - t) F (t)$ , and Mathar's recurrence then falls out by reading off the coefficient of $t^{n} / n!$ . Both steps are short. The supplementary archive includes a SymPy script that checks the ODE identically and the recurrence numerically up to $n = 500$ .

arXiv commentv2: corrected author contact email address (no math content changes)