{"id":"e-90ca7fc405d18dfa","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-32ac941c8532b0d4","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"8ac0d66ae0a0d25a0f554f903aa5c93d9118f4b2a1a41868f67ec3c537faa2c9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13236,"certificate_compressed_size":2500,"conclusion_name":"shift_x_plus_one of theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6b40f6aca669811e","role":"sequence","name":"theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","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-32ac941c8532b0d4","claim":"one_signed_coefficients"},{"node_id":"n-32ac941c8532b0d4","claim":"hadamard_kernel"},{"node_id":"n-32ac941c8532b0d4","claim":"real_rooted"}]}