{"id":"e-46cecb4caa3ed7bf","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-308cf60736924b4c","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":"498bc98ed62170aa45d749a40eda79c5211e2b0c25bbe4dd82b929fe110d73f0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12660,"certificate_compressed_size":2494,"conclusion_name":"Exponential Riordan array [x*exp(x),x].","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1c11ffc3c4cd47ed","role":"sequence","name":"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-308cf60736924b4c","claim":"real_rooted"},{"node_id":"n-308cf60736924b4c","claim":"one_signed_coefficients"},{"node_id":"n-308cf60736924b4c","claim":"hadamard_kernel"}]}