{"id":"e-0f78bcf93daf4e45","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-03cb0d97e550110a","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d25237966797de3f2eac23bd9ed178e7b3917cd252732d195f0f08fbf418e7c7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":12333,"certificate_compressed_size":2759,"conclusion_name":"obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}],"cited_by":[{"node_id":"n-03cb0d97e550110a","claim":"real_rooted"}]}