{"id":"e-c585175918782f17","operation":{"map":"A159834","operation":"basis_map"},"operation_detail":"A159834","conclusion_id":"n-f0c4b6406e0da4a3","premise_count":1,"theorem":"Hermite-type operators L f = (a + b t) f - kappa f', b > 0, kappa >= 0: every shift L - rho keeps real-rootedness (the derivative coefficient is <= 0 everywhere: Ma and Wang, EJC 15 (2008) R17, Theorem 2), and T(p) = c prod_j (L - rho_j) g for p = c prod_j (u - rho_j); so every real-rooted input has a real-rooted image","lean_route":null,"lean_declarations":[],"certificate_hash":"3b0861bef3e0391de8f68ea6e679a108436bf55baae81f0ec7cde2ac18014d8d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6410,"certificate_compressed_size":1337,"conclusion_name":"Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-f0c4b6406e0da4a3","claim":"real_rooted"}]}