{"id":"e-bc293aefd0708127","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-d4fd66ecd2fe9d3f","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":"8c3fe9df84316a158080ec0b40849ac2117a522940927a60ce7387b839755e76","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8512,"certificate_compressed_size":2315,"conclusion_name":"veronese_section_2_0 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-3272d5bc697b9cb4","role":"sequence","name":"Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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-d4fd66ecd2fe9d3f","claim":"real_rooted"},{"node_id":"n-d4fd66ecd2fe9d3f","claim":"one_signed_coefficients"},{"node_id":"n-d4fd66ecd2fe9d3f","claim":"hadamard_kernel"},{"node_id":"n-d4fd66ecd2fe9d3f","claim":"interlacing"}]}