{"id":"e-b31a65521e9db5ca","operation":{"map":"A066325","operation":"basis_map"},"operation_detail":"A066325","conclusion_id":"n-b0b0e9fb10acb2e2","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":"76b5925bbc7e2b1e038d1026fdc2790316f8c06bd756a5c2ca9e9f9d91b99b2e","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6403,"certificate_compressed_size":1325,"conclusion_name":"hermite_he","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-b0b0e9fb10acb2e2","claim":"real_rooted"}]}