{"id":"e-031667aece1ffd0c","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-9aec4822212db1dc","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"3efd5200363e02f88be5bb58ca5376d1648e65a84885a149dd9139bdd13bb3f9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2690,"certificate_compressed_size":1056,"conclusion_name":"Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[],"cited_by":[{"node_id":"n-9aec4822212db1dc","claim":"real_rooted"}]}