{"id":"e-5f70bba9f74a761e","operation":{"map":"A060821","operation":"basis_map"},"operation_detail":"A060821","conclusion_id":"n-95fc06ddb38bf951","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":"b5e854c485fcb260c4273ee077af6ef706a6969fd16c76559b276b2dbea9b4ae","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6403,"certificate_compressed_size":1331,"conclusion_name":"Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.","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-95fc06ddb38bf951","claim":"real_rooted"}]}