{"id":"e-9b675b6342ad9d47","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-bac12d13055658fe","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"4ee70252a91ce40c400b1ed2242a92f383b2a5e6a2ba858f4a7e107669100e20","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3265,"certificate_compressed_size":1106,"conclusion_name":"5-Stirling numbers of the second kind.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-bac12d13055658fe","claim":"real_rooted"},{"node_id":"n-bac12d13055658fe","claim":"one_signed_coefficients"},{"node_id":"n-bac12d13055658fe","claim":"hadamard_kernel"},{"node_id":"n-bac12d13055658fe","claim":"interlacing"}]}