{"id":"e-77b4dff64dd0f71c","operation":{"map":"A112227","operation":"basis_map"},"operation_detail":"A112227","conclusion_id":"n-0ce01d5a88917ba1","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":"1e1492af5024be4114750c5d8ee039948d1ab798f3f45cc3f878537b36eac12a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6445,"certificate_compressed_size":1347,"conclusion_name":"A scaled Hermite triangle.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-5fe240122e98ffb3","role":"sequence","name":"basis-map coefficients of A scaled Hermite triangle.","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-0ce01d5a88917ba1","claim":"real_rooted"}]}