{"id":"e-c3f8c3283ad954d0","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-1bbeb7b42d161247","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"cfb01533a2cc202c1dd0eb0064725c7fabcf5afa9feed2737e6004cf2cf38ab5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":6862,"certificate_compressed_size":2042,"conclusion_name":"veronese_section_2_1 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":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d6ec17c5486fb722","role":"sequence","name":"Triangle read by rows: row n consists of n zeros followed by n+1.","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-1bbeb7b42d161247","claim":"real_rooted"},{"node_id":"n-1bbeb7b42d161247","claim":"one_signed_coefficients"},{"node_id":"n-1bbeb7b42d161247","claim":"hadamard_kernel"}]}