{"id":"e-08a4b5e6d1540539","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-5254037deeca106e","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":"dd733c8b717c370e39f331c60c311cf1e335c9ab81c227eaf9efa05417ec9180","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":12248,"certificate_compressed_size":2713,"conclusion_name":"obreschkoff_combination of Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}],"cited_by":[{"node_id":"n-5254037deeca106e","claim":"real_rooted"}]}