Jensen Polynomials, the Charlier Identity, and an Explicit Discriminant Formula

We give a completely algebraic proof of an explicit formula for the discriminant of Jensen polynomials Jn,d(X) in the model normalisation γk = k!. The central identity — the Charlier Identity — identifies Jn,d(−nu) with a Charlier polynomial Cd (n; −1/u). Differentiating this identity gives F ′ (u) = −dn · F (n−1) (u); combined d d−1 with a contiguous relation and a resultant recurrence this yields:(A) disc(Jn,d) = H(d) · 􏰘d−2(n − k)d−k−1/n(d−1)2 , where H(d) = hyperfactorial; k=1(B) an algebraic proof that Jn,d has exactly d distinct real roots if and only if n ≥ d, recovering the stability threshold S(d) = d of Griffin–Ono–Rolen–Zagier [1] by an independent route.